Quantitative viral outgrowth assays (QVOA) use limiting dilutions of CD4+ T cells to measure the size of the latent HIV-1 reservoir, a major obstacle to curing HIV-1. Efforts to reduce the reservoir require assays that can reliably quantify its size in blood and tissues. Although QVOA is regarded as a “gold standard” for reservoir measurement, little is known about its accuracy and precision or about how cell storage conditions or laboratory-specific practices affect results. Owing to this lack of knowledge, confidence intervals around reservoir size estimates—as well as judgments of the ability of therapeutic interventions to alter the size of the replication-competent but transcriptionally inactive latent reservoir—rely on theoretical statistical assumptions about dilution assays. To address this gap, we have carried out a Bayesian statistical analysis of QVOA reliability on 75 split samples of peripheral blood mononuclear cells (PBMC) from 5 antiretroviral therapy (ART)-suppressed participants, measured using four different QVOAs at separate labs, estimating assay precision and the effect of frozen cell storage on estimated reservoir size. We found that typical assay results are expected to differ from the true value by a factor of 1.6 to 1.9 up or down. Systematic assay differences comprised a 24-fold range between the assays with highest and lowest scales, likely reflecting differences in viral outgrowth readout and input cell stimulation protocols. We also found that controlled-rate freezing and storage of samples did not cause substantial differences in QVOA compared to use of fresh cells (95% probability of < 2-fold change), supporting continued use of frozen storage to allow transport and batched analysis of samples. Finally, we simulated an early-phase clinical trial to demonstrate that batched analysis of pre- and post-therapy samples may increase power to detect a three-fold reservoir reduction by 15 to 24 percentage points.
The latent reservoir of resting CD4+ T cells is a major, if not the primary, obstacle to curing HIV. Quantitative viral outgrowth assays (QVOAs) are used to measure the latent reservoir in ART-suppressed HIV-infected people. Using QVOA is difficult, however, as the fraction of cells constituting the latent reservoir is typically about one in one million, far lower than other infectious disease biomarkers. To study reliability of these assays, we distributed 75 PBMC samples from five ART-suppressed HIV-infected participants among four labs, each conducting QVOA and following prespecified sample batching procedures. Using a Bayesian statistical method, we analyzed detailed assay output to understand how results varied within batches, between batches, and between labs. We found that, if batch variation can be controlled (i.e., a lab assays all samples in one batch), typical assay results are expected to differ from the true value by a factor of 1.6 to 1.9 up or down. We also found that freezing, storing, and thawing samples for later analysis caused no more than a 2-fold change in results. These outcomes, and the statistical methods developed to obtain them, should lead towards more precise and powerful assessments of HIV cure strategies.
Citation: Rosenbloom DIS, Bacchetti P, Stone M, Deng X, Bosch RJ, Richman DD, et al. (2019) Assessing intra-lab precision and inter-lab repeatability of outgrowth assays of HIV-1 latent reservoir size. PLoS Comput Biol 15(4): e1006849. https://doi.org/10.1371/journal.pcbi.1006849
Editor: Roland R. Regoes, ETH Zurich, SWITZERLAND
Received: April 23, 2018; Accepted: February 6, 2019; Published: April 12, 2019
Copyright: © 2019 Rosenbloom et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data and code are within the manuscript and its Supporting Information files.
Funding: A Collaboration for AIDS Vaccine Discovery (CAVD) grant from the Bill & Melinda Gates Foundation (Grant ID: 38619) supports the RAVEN Study Group. DISR was supported by amfAR Mathilde Krim Fellowship # 109511-61-RKRL. Work performed at Southern Research was supported in whole by federal funds from the NIH/NIAID/DAIDS under contract HHSN272201500017C entitled “Quantitative Viral Outgrowth Assay (QVOA) Service Resource.” DDR was supported by the NIH/NIAID Collaboratory for AIDS Research on Eradication (CARE; 1UM1AI126619) and BEAT (1UM1Al126620), the UCSD CFAR (AI306214), the Department of Veterans Affairs, and the James B. Pendleton Charitable Trust. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
The latent HIV-1 reservoir that persists following treatment with suppressive ART exists primarily in resting CD4+ T cells and is an obstacle to eradicating HIV-1 [1–4]. There are substantial ongoing efforts to eliminate or reduce the size of this reservoir [5–7]. Evaluating such efforts requires assays that can reliably quantify its size in blood and tissues in order to monitor its changes during curative intervention strategies. Replication-competent HIV-1 can be measured by QVOA. These terminal dilution assays place known numbers of resting CD4+ T cells in culture wells, usually in serial dilutions of cells that cover several orders of magnitude, with replicate wells at each dilution. The CD4+ T cells are activated before co-culture with cells that are highly permissive for primary strains of HIV. The propagation of HIV replication is detected by an assay for either p24 antigen or HIV RNA in the supernatant of these co-cultures over a two- to three-week period . Each well is read as negative or positive, which means that replication-competent virus was present in at least one of the cells in the well. The number of infectious units per million cells (IUPM) is then estimated by maximum likelihood assuming single-hit Poisson dynamics . This approach has represented the “gold standard,” because it measures replication-competent virus in latently infected cells, which is crucial because the majority of integrated HIV-1 DNA is replication-defective [10–12].
Use of QVOA presents both practical and statistical challenges, many of which are attributable to the rarity of the target entities: often on the order of only one CD4+ T cell in a million is positive for replication-competent virus by a single-round QVOA. These challenges include: 1) QVOA requires that large volumes of blood be collected to generate large numbers of Ficoll-purified peripheral blood mononuclear cells (PBMC), which are generally further processed without freezing/thawing, into input resting CD4+ T cells, 2) each assay takes weeks to complete and is expensive (∼$3,000), 3) substantial personnel time is required for cell purification, culture and monitoring supernatants for HIV replication, limiting test throughput to two to four QVOAs per lab per week, 4) not all replication-competent virus is detected by a single QVOA , and 5) different laboratories employ varying methods .
In addition, the performance characteristics of QVOAs performed within and between labs have not been carefully evaluated, a gap that this study was designed to address. The rarity of infectious units complicates the analysis of performance characteristics, because even split samples from the same collection can have large relative differences in the true numbers of infectious units that they contain. In addition, the number of infectious units in positive wells is not known, which also adds to the assays’ variability. We describe in the next section Markov-chain Monte Carlo (MCMC) methods that we developed to account for these inevitable background sources of variation while estimating additional variation, including batch effects and inter-lab variation, as well as assessing the impact of freezing PBMC samples on assay performance. We also describe simulations to validate these methods and to assess the implications of the parameter estimates. We then present the results of our method-validation simulations, estimated model parameters based on the results of four QVOAs applied to 75 split samples, and simulation results evaluating some implications of the models, before concluding with some additional discussion.
Materials and methods
Participants in the RAVEN project are enrolled and followed as part of the UCSF OPTIONS and SCOPE programs, with specific consent for apheresis collections and testing for this study as approved by the UCSF Committee on Human Research (IRB) # 10-03244.
The five ART-suppressed HIV-1 infected participants for the current study were selected to have diverse replication-competent reservoirs based on QVOA results from a previously published study . Leukapheresis collections from five HIV+ participants and one uninfected control participant were divided into 12 aliquots (control) or 15 aliquots (each HIV+ participant). Each aliquot contained roughly 300—750 million PBMC as requested by the testing labs. Three aliquots from each HIV+ participant comprised the fresh panel; all other aliquots were stored at −180° C and comprised the frozen panel.
Four labs participated in the study: University of Pittsburgh (U. Pitt.), University of California San Diego (UCSD), Johns Hopkins University (JHU), and Southern Research (SR). One aliquot per participant from the fresh panel was distributed for immediate testing to three of the labs (all except SR). The frozen panel included 18 uniquely coded liquid nitrogen-cryopreserved PBMC aliquots that were distributed to all four labs for testing 4—12 months after freezing, depending on lab testing capacity. Aliquots were blinded as to participant and aliquot identity, except that all labs had knowledge that the negative control was not included in the fresh panel. Within each lab, the frozen panel was analyzed in balanced batches of two aliquots each, designed to enable measurement of both within-batch and between-batch variation (Fig 1).
Panel a: Three frozen aliquots were provided by each HIV+ participant to each lab; an additional fresh aliquot was provided to each lab except SR. Panel b: Experimental design at one of the four labs (U. Pitt.). Fresh panel (five batches): One aliquot from each HIV+ participant was studied fresh and was not batched with any other aliquots. Frozen panel (nine batches): Three aliquots from each participant were cryopreserved and batched together with one other aliquot. Five batches contain two aliquots from the same HIV+ participant. Two HIV+ participants are chosen to supply one aliquot to the same batch (here, participants 2 and 3). The remaining three batches contain an aliquot from one of the remaining HIV+ participants and an aliquot from the negative control. In each lab, different HIV+ participants are chosen for the mixed batch (see S1 Table for complete experimental design). Panel c: Sketch of statistical model used to estimate IUPM for each participant (vi); cryopreservation effect (βs); systematic effect for each lab (βl, set to zero for U. Pitt., which was arbitrarily chosen as reference); and random variation at the level of aliquot, batch, and lab (aij, bkl, cil, respectively). These fixed and random effects combine to determine the likelihood that a given well is positive, and the likelihood of the data equals the product of likelihoods of all wells (see Eq (3)).
Labs thawed aliquots (if frozen), isolated CD4+ T cells (CD4s), and performed QVOA per lab protocol (S2 Table). S3 Table provides the well configurations (cell input counts and number of replicate wells) used for each aliquot. While QVOA output is typically reported as an estimated infection frequency and confidence interval (expressed as infectious units per million, or IUPM), all labs reported individual well outcomes (positive or negative for viral outgrowth). Reporting at this higher level of granularity allowed for more accurate statistical modeling. Individual laboratory reports were unblinded, checked for transcription errors if there were discrepancies in IUPMs calculated by the RAVEN statistical team and IUPMs reported by the laboratories, and then compiled for statistical analysis. S4 Table reports the resulting dataset.
The JHU lab used two different protocols for the fresh and frozen panels: For the fresh panel, viral outgrowth was measured at day 7 and 14 of coculture for all aliquots, but continuation to day 21 was contingent on the results at day 14. For the frozen panel, viral outgrowth was measured at days 7, 14 and 21 for all aliquots. Owing to this variance in methods, we present analysis of the data in two ways, differing in treatment of the JHU lab: Our primary analysis used cumulative QVOA results through day 21 measurements from the JHU frozen panel and no measurements from the JHU fresh panel, while our secondary analysis used day 14 measurements from both panels. As some wells require the full 21 days for outgrowth to be evident , the day 21 measurement is more sensitive, and so the primary analysis may yield more relevant characterization of QVOA precision. The secondary analysis, however, draws upon three labs instead of two for the fresh/frozen comparison and may yield more relevant characterization of the effect of cryopreservation. Unless otherwise stated, all experimental results reported draw upon the primary analysis.
As noted in the Introduction, some variability in measured IUPM is unavoidable even for a perfect assay, due to Poisson sampling variation and uncertainty about the number of infectious units that were present in positive wells. We therefore developed a statistical model that estimates additional sources of variation beyond this unavoidable background. The design permitted identification of extra variation at the aliquot level, batch-to-batch variation within each lab, and lab-to-lab variation. Below are details of the statistical model that accomplishes this task, along with our methods for fitting the model. We did not include the control participant in this modeling.
Analyzing aliquots separately.
Where a “raw” display of the data was desired (Fig 2), the following mixed effects statistical model was not used, and instead each aliquot was analyzed separately by a maximum likelihood method, as described in .
Wells in the experiment are indexed by study participant from which the sample was obtained (i = 1…5), aliquot into which the sample was divided (j = 1…15 for each participant, with 3—4 aliquots sent to each lab), assay batch (k = 1…14 for each lab), and the lab where the assay was performed (l = 1…4, referring to U. Pitt., UCSD, JHU, and SR, respectively). Viral outgrowth was assumed to follow a Poisson model, in which a well containing n input cells and response R (1 if positive for outgrowth, 0 otherwise) has likelihood (1) where Vijkl is the probability that a single cell in the well is capable of causing outgrowth. This probability is determined by the sum of fixed and random effects applying to that well (Fig 1c): (2) In this expression, Sij is set to one if aliquot j from participant i was cryopreserved before the assay, and otherwise is set to zero. Each other term represents a fixed or random effect (Table 1).
The random effects in the model (2) are normal i.i.d. variates: (4)
In this framework, σa quantifies extra variation that is distinct for each single aliquot, beyond what would be expected from Poisson sampling variability and binary readout of wells. It may be thought of as an overdispersion parameter, similar to the extra-Poisson dispersion that is present in the negative binomial distribution. The parameter σb reflects additional variation that equally influences both aliquots in the same batch, reflecting variation in laboratory personnel, condition of reagants used, or other aspects of laboratory environment. The random perturbations cil apply to all split samples from the same participant at a particular lab, but differ for the same participant at different labs. The parameter σc can therefore be thought of as reflecting participant-specific differences in exactly what each lab measures, beyond the systematic scale differences βl that apply equally to all participants. A non-zero σc indicates that each lab is measuring a quantity that correlates imperfectly with the true latent reservoir size, with smaller σc corresponding to better correlation and therefore less variation between labs in what they are actually measuring.
Maximum likelihood estimation.
We encountered difficulties fitting the above model by maximum likelihood. We did not identify any existing maximum likelihood procedures in the software packages SAS, Stata, or R that would apply directly to the above model. We therefore considered a variation in which exp(aij) follow a Gamma distribution, . This results in the number of infectious units in an aliquot having a gamma mixture of Poisson distributions, which follows a negative binomial distribution with mean dispersion parameter . To fit the resulting model using mixed effects negative binomial models, we calculated for each aliquot stand-in data rij and Nij such that a Poisson model applied to a single observation of rij with offset log(Nij) (or equivalently, including Nij as the “exposure” variable) results in the same estimated IUPM and confidence interval as the standard maximum likelihood calculations for the well-by-well data [9, 15]. With pij = IUPMij/1000000 and Sij = squared standard error of loge(IUPMij), these values are given by Nij = 1/(pij Sij) and rij = pij Nij. When no wells are positive, rij = 0 and Nij = the total number of cells in all the wells used for the assay of the ijth aliquot. We then attempted to use software for mixed effects negative binomial regression (Stata v13 menbreg command, R x64 3.4.0 glmer.nb command in the lme4 package v1.1-13, or SAS v9.4 proc Glimmix) to fit the needed model to the rij and Nij, with crossed random effects bkl and cil.
All the software packages failed to produce satisfactory results when applied to our data (see “Simulation likelihood estimation underperformed MCMC”). The difficulty of reliably fitting the crossed random effects model precluded application of maximum likelihood to simulations of our full study (“Validation by simulation: Multiple labs”). In addition, simulation results for the simplified situation of a single lab (“Validation by simulation: Single lab”) were not fully satisfactory, with substantial bias toward random effect variances of zero and poor coverage probabilities. For example, in the large variation case, more than half the estimates of σb were zero and profile likelihood 95% confidence intervals covered the true value only 73% of the time. Because of these difficulties and poor performance, we focused primarily on Bayesian estimation as described in the next section.
Markov-chain Monte Carlo estimation.
Given the difficulties with maximum likelihood estimation and the versatility of Markov-chain Monte Carlo (MCMC) approaches, we developed a Bayesian estimation framework. In order to obtain posterior distributions that mainly reflect the evidence provided by our data, we aimed to use prior distribution assumptions for the parameters that were “weak.” To make our assumptions neutral about the existence of nonzero random effects, we used prior distributions for each of σa, σb, and σc with 50% point mass probability at zero.
A uniform or normal distribution was used as prior for each fixed-effect parameter, and a mixture of a point-mass at zero and a half-Cauchy prior was used for each random effect parameter; this prior has a heavy tail, allowing the model to explore large values of σ [16, 17], while also allowing it to take on the null value of no extra variation at all: (5) where half-Cauchy scale parameters τ = 1, 2, 3, 4 (corresponding to the median of the distribution) were tried.
Posterior distributions for parameters were estimated using PyStan 2.12 . As this program does not permit direct use of priors with point mass at zero, the implementation used an ensemble model constructed from eight separate components, one for each combination of the three random effects in Table 1. The posterior weight for each model component was calculated using the Watanabe-Akaike Information Criterion (WAIC) . This ensemble therefore achieves the prior distributions in Eq (5) (see , Section 6.5.2, “Model averaging”). Complete Stan model files are provided in S1 Appendix.
For each model component, four parallel chains of 2500 iterations each were sampled. The first half of each chain was discarded as warmup. To ensure a valid fit, trace plots of the chains were visually inspected for mixing, and diagnostics Neff (effective sample size) > 200 and (scale reduction factor) < 1.1 were required; in most cases, diagnostics well exceeded these criteria.
A set of 8000 independent samples from the posterior parameter distribution of the ensemble model was then obtained by combining 1000 samples from the posterior of each component. Using the component posterior weights, weighted quantiles of these 8000 samples were computed. Posterior medians and 95% credible intervals (spanning 2.5 to 97.5 percentiles) were reported. Unless otherwise noted, all reported results use these weighted ensemble estimates. Where a sample from the joint posterior was required for simulation, 1000 of the 8000 samples were selected, roughly proportional to the component weights (“complete ensemble”). An alternate ensemble of 1000 was also constructed setting batch effect to zero; in this case only four of the eight model components were included (“batch variation-free ensemble”). Finally, to provide more traditional estimates of fixed effects, posterior medians and 95% credible intervals were also reported for the model component excluding all excess variation.
Most estimates are reported as fold-changes, that is, the exponential of a model parameter. Random variation at the aliquot and batch levels combined is reported as ; variation at all three levels combined is reported as .
Validation by simulation: Single lab.
For single-lab simulations ignoring cryopreservation, the five participants’ IUPMs were based on preliminary estimates from fresh samples: 0.104, 0.591, 0.654, 0.835, 1.454. Three scenarios of extra-Poisson variation were considered: Large variation (σa = σb = 0.7), moderate variation (σa = σb = 0.2), and no variation (σa = σb = 0). The large and moderate variation scenarios were chosen to bracket the range of preliminary (maximum-likelihood) estimates from the data. The simulated assay used six replicate wells of cell inputs 1,000,000, 300,000, 100,000, 30,000, 10,000. Nine batches were assigned for each simulation: six batches with two aliquots each and three batches with a single aliquot each. Two aliquots from each participant were assigned to the same batch, one aliquot from each of two randomly chosen participants were assigned to a batch together, and the remaining three aliquots (each from a separate participant) were assigned to the singleton batches. A separate random choice of participants for the mixed batch was made for each simulation.
In the scenario with no variation, each well is positive with probability 1 − exp(−n exp(vi)), where n is the number of input cells and vi is the log of infection frequency for participant i. In each scenario with variation, two different formulations were tested. Normal formulation: Random effects aij and bkl were sampled from normal distributions (4), and each well is positive with probability 1 − exp(−n exp(vi + aij + bkl)). Gamma formulation: Batch random effect bkl was sampled as in the normal formulation, and each well is positive with probability , representing the probability that a negative binomial random variable is positive when it has mean n exp(vi + bkl) and dispersion parameter .
Validation by simulation: Multiple labs.
For multi-lab simulations including both fresh and cryopreserved samples, the full experiment as specified in S3 Table was simulated, using ensemble posterior median parameter estimates reported in S10 Table (“All four labs, using JHU day 21 readout”). On the natural log scale, these estimates are σa = 0.40, σb = 0.57, σc = 0.44, βS = −0.38, β2 = 2.22, β3 = −0.22, and β4 = −0.93. These estimates were obtained using the prior, which was putatively chosen after preliminary analysis indicated that both σa and σb were less than 1 (the median of the distribution). The multi-lab simulation was then used to compare performance of to other priors and check the consistency of the MCMC method. Random effects and the probability that each well is positive followed the model described above (“Statistical model”).
In addition, to compute the probability that the method may overestimate σa, σb, or σc despite the true parameter value equaling zero, we repeated the simulations three times, each time replacing one of these three parameters with zero. We then reported the average probability of overestimation among these simulations.
2000 simulations of each specification were performed, except for the analysis considering true parameter values of zero, for which only 100 were performed. ML estimation (above, “Maximum likelihood estimation”) was used for data resulting from single-lab simulations, and MCMC estimation (above, “Markov-chain Monte Carlo estimation”) was used for both single- and multi-lab simulations. For a simulated true parameter θ and distribution of median posterior estimates , accuracy of the MCMC method was characterized both as bias (median and middle 95% of ) and as absolute error (median of ); these values were divided by ln(10), i.e., reported as log10-error.
Characterizing accuracy of assays.
Simulations were used to illustrate the typical accuracy of each lab’s assay in the absence of batch effects (assuming that these could be rendered irrelevant by assaying specimens from the same person—such as baseline and post-treatment—in the same batch). For each of 1000 samples from the batch effect-free ensemble posterior (see above, “Markov-chain Monte Carlo estimation”) and each assumed true IUPM, 1000 simulations of well-by-well output were performed for each assay, each with a separate random draw of aliquot-level and lab-level random effects. In each simulation, the maximum likelihood estimate of IUPM was compared to the true IUPM (scaled by lab-specific factor βl) to obtain the log10-error. The median of absolute log10-error was then recorded for each of the 1000 samples from the ensemble posterior. For each lab’s assay, we then report the median, 2.5%ile, and 97.5%ile of these median absolute errors across the posterior sample.
We ran two versions of these simulations. For the first version, we supposed that each assay has equal claim to biological truth and that lab-based random effects cil are discrepancies from a consensus standard. These random effects were drawn from , where σc is the estimated value drawn from the ensemble posterior. For the second version, we chose the JHU assay (index l = 3) to represent the standard against which other assays are measured. Under this assumption, the differences cil − ci,3 are lab l’s discrepancies from the standard. To simulate data in which the cil themselves are the discrepancies, we sampled these random effects from for all l ≠ 3. All JHU lab-based random effects ci,3 were set to zero.
The same simulations were also used to compare accuracy between each pair of assays. For each assumed true IUPM and ensemble posterior sample, the difference in median absolute log10-error between two assays was recorded for the 1000 matched simulations. For each pair of assays, we report the median, 2.5%ile, and 97.5%ile of these differences across the posterior sample.
At JHU and SR, the number of replicate wells with one million cells apiece varied with the number of rCD4s purified from each aliquot. First, to mimic typical aliquots, we simulated the JHU assay with 26 replicates and the SR assay with 12 replicates. Second, to consider the situation where JHU and SR used a number of cells similar to those used by U. Pitt. and UCSD, we simulated both assays with 8 replicates.
Simulation of clinical trial.
Latency-reducing treatment was simulated to illustrate the power and accuracy of QVOA for measuring changes in IUPM. Pretreatment log10 IUPM from each participant was assumed to follow a normal distribution with mean −0.013 and standard deviation 0.675, consistent with the cohort studied in . Within-person change in log10 IUPM from pre- to post-treatment was normally distributed with mean δ and standard deviation σδ. In control participants, “pre” and “post” IUPMs were equal. Both JHU and UCSD assay protocols were simulated, using ensemble posterior median parameter estimates reported in S10 Table (“All four labs, using JHU day 21 readout”). For the JHU assay, we simulated 50 million-cell wells for fresh samples, 26 million-cell wells for frozen samples, and two wells of each other size (five-fold dilution from 200,000 to 320). The UCSD assay was simulated according to the configuration in S3 Table.
Both batched and unbatched analyses were studied. In batched analysis, all samples were treated as cryopreserved, and only aliquot- and lab-level excess variation were included (all batch effects bkl = 0). In unbatched analysis, all samples were treated as fresh, and all three levels of excess variation were included. In both cases, the same lab-level random effect cil was assumed to hold for both samples from each participant. For each sample, the maximum likelihood IUPM was estimated, with negative assay results replaced with a value equal to one-half of the minimum possible positive assay value. A before-after change in log10 IUPM was calculated for each individual from these IUPM estimates.
Two treatment effects were studied: a 10-fold median reduction in IUPM (strong effect), but with substantial person-to-person variation (δ = −1, σδ = 0.5), and a weak effect with lower person-to-person variation (δ = −log10(3) ≈ −0.48, σδ = 0.16). We intended the strong effect to represent a treatment with clinical potential, and we simulated an early, discovery-oriented study with 6 treated participants and no controls, analyzed by a paired t-test. We intended the weak effect to represent a treatment with minor improvement over those currently reported , of interest for providing information about the biology of HIV or as a possible component of future combination treatments. We assumed greater concern in this situation about background change over time and assay drift, so we simulated studies with 12 treated and 12 control individuals, with change in log10 IUPM analyzed by unpaired t-test. We also evaluated an alternative approach to analysis of the simulated data. This approach used maximum likelihood applied to all the well-level assay results to estimate a model with fixed effects for treatment and each person’s baseline IUPM, along with a random effect for within-person change. (For the weak effect scenario, different random effect variances were allowed for treated versus control participants.) 1000 simulations were analyzed for each scenario, for 8000 simulations in all (JHU/UCSD assay protocols, batched/unbatched analyses, weak/strong treatment effect). Power was calculated as the fraction of simulations for which p < 0.05 in the t-test or maximum likelihood analysis.
Simulation validates MCMC fitting of ensemble model
We first tested the MCMC method in simulations of the frozen panel at a single lab (Methods: “Validation by simulation: Single lab”). We found that fitting a model that forces aliquot- and batch-level random effects to be positive led to over-estimates of these effects, particularly when the simulated effect was moderate or nonexistent (median biases up to +0.132 log 10, S5 Table). Analysis by maximum likelihood (Methods: “Maximum likelihood estimation”) gave poorer results. These findings indicated that priors for the random effect variances would be suitable only if they included mass at zero. We therefore implemented priors with 50% point masses at zero by using MCMC to fit an ensemble of eight separate model components, together allowing for the eight possible combinations of presence/absence of the three random effects (aliquot-level, batch-level, and lab-level, see Methods: “Markov-chain Monte Carlo estimation”).
In simulations mimicking the 4-lab experimental design (Methods: “Validation by simulation: Multiple labs”), we found that MCMC fits of the ensemble model estimated all fixed effects and the between-lab random effect very well. Median biases in these estimates were between −0.013 and +0.012 log 10, and 95% CI coverage was between 93.2% and 95.0% (Table 2). The model produced similar estimates for all priors tested. The prior did slightly outperform the other three priors tested (S6, S7 and S8 Tables), and so it was chosen for all analyses of experimental data.
The experiment has a total of 194 million to 289 million cells from each of five participants, distributed among four labs, encompassing 474 to 569 wells per participant. Bias and absolute error represent over- or underestimation of effect sizes, as difference in log10 of fold-change. For example, typical batch effects are estimated to be 10−0.202 = 63% of the simulated truth. See Methods (“Validation by simulation: Multiple labs”) for simulation details.
The method also estimated the combined effect of aliquot-level and batch-level variation well (median bias −0.024 log 10, 95% CI coverage 96.6%). Yet it was more difficult to disentangle variation between these two levels: Aliquot-level variation was overestimated slightly (median bias +0.043 log 10), batch-level variation was substantially underestimated (median bias −0.202 log 10), and both had coverage slightly lower than expected. This difficulty likely arises from experimental constraints that limited batches to only one or two aliquots each. This mis-assignment of batch-level variation to the aliquot level may also explain bias and low coverage for the combined effect of aliquot-level and lab-level variation. An experimental design with larger batches may be less vulnerable to these issues.
To investigate whether MCMC fitting of the ensemble model might lead to a false conclusion that variation is present at a given level, we simulated data for which one or more of the three sources of variation was removed. When all three sources were removed from simulations, MCMC rarely produced estimates surpassing the level of variation observed in the experimental data (posterior probability weight typically no more than 0.1%, bottom row of S9 Table). When one or two sources of variation were included, MCMC often misidentified the source of variation, though misidentification as aliquot-level variation was less common and typically produced estimates smaller than the experimental estimate. Even when one source was misidentified as another, estimates of total variation coming from all three levels performed well.
MCMC method estimates systematic and random variation in QVOA
Although the goal of each lab’s assay is the same—to quantify infectious provirus infecting resting CD4+ T cells—infection levels reported by UCSD were consistently higher than those reported by the other three labs (Fig 2). We used the Bayesian model to estimate the systematic effect of assay characteristics and lab practices, measured as fold-change from U. Pitt. as reference (Table 3). When accounting for differing cell counts in each assay, as well as excess random variation at the aliquot, batch, and lab levels, we found that UCSD reported IUPMs averaging 9.2-fold higher than those reported by U. Pitt. (95% CI 3.8—24). For the other two labs, credible intervals for this systematic effect spanned 1. This result is not surprising, given methodological differences in each assay. U. Pitt., JHU, and SR each recorded a well as positive for viral outgrowth if levels of viral protein p24 measured by ELISA exceeded that of a threshold reference sample. UCSD, on the other hand, used viral RNA detection. While RNA detection is more sensitive than the p24 assay , it may have greater potential for false positive results. In fact, two wells of a single negative coded control aliquot were reported as positive (Fig 2).
Posterior medians and 95% CIs shown.
Both the systematic effect and assay input cell counts determine the frequency of all-negative assay results. While SR’s experimental protocol was modeled on that of JHU (S2 Table), they reported more all-negative aliquots (4 of 15 versus 1 of 15 aliquots for JHU). This observation reflects the fact that SR generally recovered fewer resting CD4s from each aliquot than JHU did for input into QVOA (average of 12.7 versus 26.8 million cells per aliquot, S3 Table), and it might also reflect a systematically lower IUPM scale (half that of JHU, although credible intervals overlap).
In addition to this systematic lab effect, we determined that random variation in excess of the baseline Poisson-binomial model was likely (posterior probability 0.85) at all three levels—between-aliquot, between-batch, and between-lab. Excess variation in at least one level was a near-certainty (posterior probability = 1 − 10−22). Table 4 summarizes estimates of excess variation at each level. As study design was limited to batches of only one or two aliquots, it is not easy to disentangle variation at the aliquot and batch levels; estimates are inversely correlated with one another (slope −0.69, S1 Fig). Combining variation at both of these levels, we estimate that two aliquots, studied in two different batches at the same lab, are expected to vary 2.0-fold in excess of Poisson variation (95% CI 1.6—2.7). Two aliquots, studied at two different labs, are expected to vary 2.3-fold in excess of Poisson variation (95% CI 1.8—3.5), an estimate obtained by combining variation at all three levels.
Cryopreservation did not have a major effect on QVOA outcomes
When accounting for excess variation, all credible intervals estimated for the effect of cryopreservation spanned 1-fold change, or the absence of an effect (Table 5). When using results from all three labs that tested fresh and frozen panel aliquots (SR did not test fresh PBMC samples) to estimate a single effect size, we estimated between 0.56- and 1.97-fold change in infection frequency compared to fresh aliquots. In the context of the large (> 100-fold) reductions sought by latency-reducing therapies, this effect is not major. Credible intervals were wider when the effect was estimated for each lab separately, and the interval was particularly wide for JHU (25-fold difference between top and bottom of interval), owing to the larger number of all-negative aliquots at this lab.
Precision within and across labs
Running MCMC analysis on each lab separately suggested that each lab’s assay offered a similar level of precision. The median estimates for aliquot- and batch-level variation for each lab fell within the 95% credible interval of the joint estimates in Table 4, and none were more than 21% away from the corresponding median estimate (S10 Table, S2 Fig). Precision of early readout (day 14) from JHU assays was, however, estimated to be lower than that of the other assays: There was combined 3.5-fold variation at both levels (95% CI 2.0- to 10.7-fold), which is 69% higher (95% CI 10% smaller to 458% higher) than the joint lab estimate that included data from the later JHU readout. This difference in precision may reflect the fact that allowing more time for exponential growth leads to a stronger p24 signal and clearer distinctions between positive and negative wells. S11 Table provides full joint posteriors for each separate analysis.
Neglecting the strong evidence supporting excess random variation at multiple levels (Table 4) can generate misleading interpretations about precision of lab procedures. To demonstrate the relevance of accounting for excess random variation, we recomputed all estimates in a model excluding this excess variation (Tables 3 and 5, right columns). While median estimates did not change greatly, credible intervals shrank by 2- to 10-fold for each parameter. Neglecting this excess variation therefore overstates certainty in parameter estimates. One particular effect of this error in our experiment would be to conclude, rather strongly (p < 0.001), that cryopreservation increases observed infection frequencies in the JHU lab.
Paying close attention to the sources of random and systematic variation can help in choosing assays for and optimizing design of clinical trials for latency-reducing therapies. In the next two sections, we demonstrate how simulations based on the parameter estimates described above may guide this effort.
Simulations based on experimental outcomes can be used to compare assay accuracy
While each lab’s assay aims to measure the replication-competent HIV latent reservoir, protocols differ among them: U. Pitt., JHU, and SR use a p24 antigen test to detect viral outgrowth, while UCSD uses an RNA PCR test; U. Pitt., JHU, and SR use PHA and gamma-irradiated PBMCs to stimulate resting cells, while UCSD uses antibody to CD3/CD28 bound to the culture plate (S2 Table). Additionally, target cells added to propagate virus differ among labs. These protocol differences may explain both systematic and random variation between labs (Tables 3 and 4). We may think of these labs as measuring different aspects of latency, each with a valid claim to being a meaningful measure, with experimental and biological motivations for specific protocol choices.
In the absence of an external standard defining latent reservoir size, we can nonetheless address how sensitivity of an assay affects its accuracy. Specifically, assays that use more input cells overall or have systematically high IUPMs (high fold-change in Table 3) will have improved sensitivity. By drawing from the joint posterior distribution (S11 Table) and simulating data for each draw, we investigated how sensitivity relates to the accuracy of measuring small reservoirs (Methods: “Characterizing accuracy of assays”).
For typical infection frequencies (1 IUPM on the U. Pitt. scale, higher or lower for the other assays according to systematic effect, Table 3), all four assays have nearly identical accuracy, as measured by median absolute error from a consensus standard (Fig 3). As infection frequency declines from this level, error increases for all four assays, most sharply for SR, but gradually for UCSD and JHU. In the case of SR simulated at 0.1 IUPM on the U. Pitt. scale (median of 0.039 IUPM on the SR scale, Table 3), for a majority of parameter draws, a majority of simulated assay outcomes are all-negative, resulting in infinite error on the log scale. In the case of UCSD and JHU, improved accuracy has costs: The more sensitive RNA-based assay used by UCSD may have a higher false positive rate (2 of 18 control wells with one million cells were reported as positive, compared to none of the 193 control wells with one million cells among the other three assays, S4 Table), and JHU exhausted a larger sample (26.5 million cells simulated, versus 8.6 to 12.5 million for the other assays). If the JHU assay is simulated with fewer cells, bringing it in line with the other assays, its accuracy profile lies between that of U. Pitt and SR (S3 Fig); overlapping credible intervals in accuracy reflect overlapping credible intervals in systematic lab effects.
Each assay is measured against a consensus standard, appropriately scaled by βl for that assay. “All-negative” represents infinite error on the fold-change scale, which occurs when the maximum likelihood estimate of IUPM is zero. Median estimate and 95% credible intervals shown for 0.1, 0.2, 0.5, 1, 2, and 4 IUPM on the U. Pitt. scale. At IUPMs of 1 or more, measured values in these assays are expected to be within 1.6- to 1.9-fold of the truth.
If false positives did in fact occur in HIV+ samples studied by the UCSD RNA-based assay, it would be reflected in our estimates as both improved sensitivity for UCSD (higher β2) and reduced correlation between labs (higher σc). We used simulations to investigate the possible tradeoff between sensitivity and specificity, treating the JHU assay as a gold standard (not subject to lab-based random effect) and subjecting UCSD to the total between-lab variation (applying a random effect with standard deviation to each participant). If an equal number of cells is made available to both assays, then the extra sensitivity of RNA readout outweighs the cost of extra variability for particularly small reservoirs (IUPM of 0.2 or less on the U. Pitt. scale, Fig 4). Uncertainty in estimates of between-lab variation, UCSD systematic effect, and JHU systematic effect, however, make definitive comparisons difficult; credible intervals for the UCSD/JHU accuracy difference overlap zero for all IUPMs simulated. If the JHU assay instead uses roughly three times as many cells as UCSD, as in the experimental study, then it does not suffer the same disadvantage at the low IUPMs simulated and enjoys a somewhat larger advantage at higher IUPMs. The accuracy of any two assays may be compared head-to-head by the same simulation method, judging their performance against a consensus standard (applying σc equally to both assays, S12 and S13 Tables) or against a chosen standard assay (applying to other assays, S14 and S15 Tables).
Batch variation-free ensemble estimates of parameters were used in simulations. Median estimate and 95% credible intervals shown for 0.1, 0.2, 0.5, 1, and 2 IUPM on the U. Pitt. scale. All values plotted are also provided in S14 and S15 Tables.
Batched study design can improve power to detect latency reduction
Cryopreserving specimens allows samples taken at different points in time to be thawed and analyzed together in the same batch. This strategy eliminates the estimated 1.8-fold batch variation that would otherwise affect longitudinal comparisons in clinical trials, potentially boosting power to detect reduction in latency. To investigate this possibility, we simulated and analyzed data for two hypothetical latency-reducing therapies with strong effect (10-fold reduction in latency) and weak effect (3-fold reduction) (see Methods: “Simulation of clinical trial”). For each therapy, we simulated both a p24 assay based on JHU protocol, which uses all available cells from a participant, and an RNA assay based on UCSD protocol, which uses a fixed number of cells regardless of availability. Consistent with JHU assay cell counts (S3 Table), we supposed that roughly half as many cells would be recovered from cryopreservation as would be available fresh.
For the weaker therapy, batching substantially improved power to detect and accuracy to measure latency reduction following treatment (Table 6). Improvement was greater for the UCSD protocol, with power increasing from 58.6% to 82.6% and median absolute error declining from 0.14 log10 to 0.10 log10. Although freezing-thawing sacrificed half of the resting cells available to the JHU protocol, the benefits of batching overcame this deficit, leading to a 15 percentage-point increase in study power.
For the stronger therapy studied in a smaller cohort, batching provided a modest benefit to the UCSD protocol, but reduced cell availability in the JHU protocol eliminated this benefit completely. Moreover, both protocols underestimated the effect of treatment (−7% to −34% bias, Table 6). This outcome resulted from our use of an imputed IUPM in the case of negative assay results, a common practice in the field . Strong therapy that reduces latency 10-fold can often result in true IUPMs well below the imputed value. In the batched JHU protocol, 35% of samples returned a negative assay, leading to the largest underestimate of treatment effect. To provide an alternative to the t-test that avoided imputation, a maximum likelihood estimate of effect size was also computed. When this method was used, batching improved both power and accuracy in all cases, but the resulting confidence intervals had poorer coverage (S16 Table).
Maximum likelihood estimation underperformed MCMC
None of the three software packages that we tried produced fully satisfactory results when applied to the experimental data (Methods: “Maximum likelihood estimation”). The best log-likelihood obtained was from a model in Stata that excluded between-lab variation. When this variation was included, the log-likelihood closely approached that of the best model before ending with an error. In SAS, the model fitting completed for essentially the same model, with σc = 0, but with most parameters having standard errors equal to zero. In R, the model converged to substantially different parameter values, with a much worse log-likelihood.
Aside from the lack of between-lab variation, which may reflect the bias toward zero seen in single-lab simulations (see Methods: Maximum likelihood estimation), many results by maximum likelihood were similar to the MCMC results given above. The cryopreservation effect and systematic lab effects were all within 10% of MCMC posterior medians. The typical between-aliquot excess variation was estimated to be 1.61-fold (vs 1.50-fold by MCMC), with a p-value of 0.0003. The typical between-batch excess variation was estimated to be 1.43-fold (vs 1.77-fold by MCMC), with p = 0.098. A likelihood ratio test for at least one source of excess variation had p = 1.4 × 10−13.
Rigorous evaluation of the performance of QVOAs is challenging due to their high cost and the presence of unavoidable background variation. By applying detailed statistical analysis to 75 split PBMC samples from 5 ART-suppressed HIV+donors, we found strong evidence for additional variation beyond the unavoidable background, even within a single batch at a single lab. When a donor’s true IUPM is 1 or more (on the U. Pitt. scale; Table 3), we estimated that assay results typically differ from the true IUPM by a factor of only 1.6 to 1.9 up or down (Fig 3). At lower true IUPMs, some assays had much larger median errors, while others avoided major increases by use of a large number of input cells (JHU assay) or by use of a readout that was more often positive (UCSD assay). We caution that these findings depend both on assay configuration (number of wells of each input cell number) and consistency of experimental protocols. In this study, QVOA was conducted by four labs with over 50 years’ combined experience developing and performing this labor-intensive assay; we do not know whether QVOA performance may differ in other lab settings.
We found that the four assays appeared to differ from each other both systematically (in scale; see Table 3) and randomly (between-lab variation in Table 4). These differences may reflect the differing subsets of infectious units (IU) that the assays detect. Recent work suggests that QVOAs detect only a fraction of the IU present in a sample . The assays’ different stimulation and detection procedures likely cause each to measure a different subset of IU. Different inducers of T cell activation have variable effectiveness. In one study, the use of antibodies to CD3 and CD28 was more effective than other methods . For the detection step, HIV RNA by PCR is more sensitive than digital ELISA assays for p24 antigen (e.g., Simoa ), which in turn is more sensitive than the standard ELISA assay for p24 antigen. Details of CD4 cell separation and aspects of cell co-culture details like cell type and medium could also create variability between labs. Measuring a larger subset of IU is desirable, because sampling variation is relatively smaller when counts are higher, reducing relative error. A caveat, however, is that the fraction of the total latent reservoir within a smaller subset could, at least in principle, be more stable within and between individuals than that within a larger subset. We could not assess this possibility, because we had no gold standard measurement of the entire reservoir. In addition, higher readings that are due to false positive wells will not improve measurement. The UCSD assay, which used an in-house RT-PCR assay to detect HIV RNA rather than p24 antigen in culture supernatants, had two false positive wells among 126 wells evaluated for an HIV-uninfected donor. These were contiguous wells with one million cells each within a single one of three split samples. This may have been due to contamination during culture supernatant testing by RT-PCR, and in any case would not be a high enough rate to account for much of the UCSD scale factor.
The existence of substantial measurement error in QVOA has implications for the design of clinical studies that use QVOA to evaluate eradication strategies. Classical sample size calculations depend on the size of the latency reduction effect and how much it varies from person to person (δ and σδ in the simulated clinical trials described above). Measurement error increases the effective person-to-person variation, which increases the sample size needed for any given level of desired precision. We do not, however, recommend assuming that measurement error is the only source of person-to-person variation in the observed effect. Interventions may well have truly differing effects for different people, and this variation may be as large or larger than the variation due to measurement error. Unfortunately, the variation in intervention effects may be difficult to accurately anticipate at the planning stage of a study .
Another implication is the need to use designs that will not be biased by regression-to-the-mean phenomena. For example, it might seem reasonable to exclude from an intervention study potential participants who have QVOA results that are all negative (measured IUPM = 0) at baseline, because they cannot go any lower. This exclusion, however, will tend to cause the remaining participants to average lower on repeat testing than at baseline, because the remaining subset has been depleted of some downward measurement errors by the exclusion, while the repeat testing will include an unbiased array of measurement errors. Therefore, a randomized parallel design should be used if such a baseline exclusion is applied; treated and control arms will then have comparable regression to the mean, resulting in a fair assessment of the intervention effect.
We found evidence that controlled-rate freezing and liquid nitrogen storage and shipping of PBMC did not cause substantial differences in QVOA results compared to use of fresh cells. This is encouraging because of the practical difficulties in using fresh cells and because storage enables assaying multiple specimens from the same person in the same batch, eliminating batch variation from estimates of within-person changes. It also permits clinical trials to be more readily conducted in trial sites without local laboratories qualified to perform reservoir assays. Our simulations showed the potential for gains in power and precision from such batching. There are, however, some important caveats regarding storage. We had only 15 results on fresh cells, with no split samples evaluated at the same lab. Despite this, our CI’s were narrow enough to provide good evidence against any systematic change due to storage of 2-fold or more (Table 5). In addition, the lack of split fresh samples precluded effective evaluation of the effect of storage on measurement variability. We also did not evaluate the effect of storage duration. The SR assay was performed later, after longer storage, than the other assays, and it also tended to have the lowest IUPMs. Using similar protocols to the JHU assay, it also obtained lower numbers of input cells, and JHU obtained lower input from stored samples than from the fresh samples collected at the same time. Consequently, until additonal studies on impact of duration of storage on QVOA IUPMs are conducted, we recommend that serial PBMC samples from participants in intervention studies be frozen and batch tested as soon as practical following collection.
Our statistical methods and results indicate potential value for additional statistical research on how to estimate and analyze QVOA data (and data from similar limiting dilution assays). We found a 98% posterior probability of extra-Poisson variation between split samples in the same batch at the same lab (Table 4). This suggests that there could also be extra-Poisson variability between wells within a single assay run, which would violate the single-hit Poisson dynamics that are typically assumed when calculating IUPM. A modified approach might be warranted in which the usual Poisson formulas are replaced with formulas based on the negative binomial distribution, which allows for overdispersion. A reasonable choice of overdispersion parameter would be our posterior median estimated of about 0.16, or it could be estimated as an auxiliary parameter of the calculation. Alternatively, the usual maximum likelihood approach could be replaced with a Bayesian posterior median. For analysis of the full 75 QVOA results in our study, we found that Bayesian MCMC estimation outperformed maximum likelihood. Although our study differs from clinical studies of interventions to reduce the latent reservoir, there may be potential for MCMC estimation to improve the analysis of clinical studies, too.
This study provides evidence that QVOA assays have extra variation, beyond what is theoretically inevitable, at three levels: between split samples even in the same batch, between batches run using the same assay, and between different assays. Results for stored frozen samples did not appear to differ systematically from those on fresh samples. We developed and validated methods for fitting detailed statistical models of assay variation. We are now using these methods to evaluate faster and cheaper alternatives to the classical QVOA assays evaluated here .
S1 Table. Experimental setup, showing lab and batch assignment of each aliquot.
S5 Table. Performance of MCMC estimation using the model with positive aliquot- and batch-level variation and prior, in single-lab simulation.
S6 Table. Performance of MCMC estimation using the ensemble model and prior, in multi-lab simulation, as in Table 2.
S7 Table. Performance of MCMC estimation using the ensemble model and prior, in multi-lab simulation, as in Table 2.
S8 Table. Performance of MCMC estimation using the ensemble model and prior, in multi-lab simulation, as in Table 2.
S9 Table. Performance of MCMC estimation assuming a true random effect parameter of zero, using prior, in multi-lab simulation.
S10 Table. Summary of model fits.
Parameter distributions of the ensemble model are reported as weighted quantiles of 8000 posterior samples, and the number of samples in the discrete ensembles are shown (see Methods: “Markov-chain Monte Carlo estimation”). Analysis is shown for eight different subsets of the data: three subsets including multiple labs, and five subsets including a single lab each.
S11 Table. Full joint posteriors for model fits and membership in discrete ensemble posteriors.
Both discrete ensemble posteriors are provided for each subset of the data, each containing 1000 samples (see Methods: “Markov-chain Monte Carlo estimation”). Quantiles of both discrete ensembles are shown. Additionally, quantiles are shown for the single model estimating all three random effects.
S12 Table. Pairwise comparisons of assay accuracy for lower IUPMs, assuming that each assay has equal claim to biological truth (σc applied to both assays).
Batch variation-free ensemble estimates of parameters were used in simulations. Each entry shows median improvement of row versus column assay (median and 95% CI sampling from ensemble parameter distribution). IUPMs are shown on the U. Pitt. scale. Darker shading: Entire CI is above zero (green) or below zero (red). Lighter shading: Median estimate is above 0.05 (> 12% increase, green) or below −0.05 (> 11% decrease, red), but the CI crosses zero. “Infinite” difference in accuracy indicates that a majority of simulations of the disfavored assay have all-negative outcomes (maximum likelihood estimate of zero). Assay configurations match those in the experimental study, except “JHU (8M)” and “SR (8M),” which use fewer wells to match the cell input count of U. Pitt. and UCSD assays.
S13 Table. Pairwise comparisons of assay accuracy for higher IUPMs, assuming that each assay has equal claim to biological truth (σc applied to both assays).
Method and format match S12 Table.
S14 Table. Pairwise comparisons of assay accuracy for lower IUPMs, treating the JHU assay as gold standard ( applied to other assays).
Method and format match S12 Table.
S15 Table. Pairwise comparisons of assay accuracy for higher IUPMs, treating the JHU assay as gold standard ( applied to other assays).
Method and format match S12 Table.
S16 Table. Simulated results of latency reduction trials, maximum likelihood estimation to compare pre- and post-treatment data.
S1 Fig. Scatterplot of estimates of excess variation at the aliquot, batch, and lab levels, expressed as the natural log of fold-change (parameters σa, σb, σc).
Panels show robust linear regression and Wald test p-values (null model = zero slope) of 1000 samples from the ensemble posterior.
S2 Fig. Sensitivity analysis of of aliquot-level, batch-level, and lab-level variation.
S3 Fig. Accuracy of assays used in the experimental study, with JHU and SR assays modified to use the same number of input cells as U. Pitt. and UCSD.
Each assay is measured against a consensus standard, appropriately scaled by βl for that assay. “All-negative” on the y-axis represents infinite error on the fold-change scale, which occurs when the maximum likelihood estimate of IUPM is zero. Bars show 95% credible intervals around median posterior estimates.
S1 Appendix. Stan model code.
Complete model specifications are provided in the Stan programming language (version 2.12) as supplementary text files. Each filename has a binary code, indicating presence/absence of aliquot and batch effects (for single-lab models) or presence/absence of aliquot, batch, and lab effects (for multi-lab models). In the code, SourceID refers to participant index i, UniqAliquot refers to concatenated participant-aliquot index ij, UniqBatch refers to concatenated batch-lab index kl, AssaySourceID refers to concatenated participant-lab index il, Assay refers to lab index l, and Stored refers to cryopreservation indicator Sij. Note that all effects are represented as logit values, ensuring that all frequencies remain between 0 and 1 as MCMC explores the parameter space.
Members of the RAVEN Study Group: Steering Committee: Michael P. Busch, Silvija Staprans, John Mellors, Doug Richman, Steve Deeks, Peter Bacchetti, Joe Fitzgibbon. Central Lab Staff: Mars Stone, Sheila Keating, Sonia Bakkour, Xutao Deng. Key Collaborators: Ron Bosch, Rebecca Ho, Robert Siliciano, Janet Siliciano, Nicolas Chomont, Daniel Rosenbloom, Roger Ptak, Sulggi Lee.
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