Andreas Markoulidakis, Matthew Hickman, Nicky J Welton, Loukia Meligkotsidou, Hayley E Jones
10 min
Abstract
Estimating the number of the number of people from hidden and/or marginalised populations - such as people dependent on opioids or cocaine - is important to guide policy decisions and provision of harm reduction services. Methods such as capture-recapture are widely used, but rely on assumptions that are often violated and not feasible in specific applications. We describe a Bayesian modelling approach called Multi-Parameter Estimation of Prevalence (MPEP). The MPEP approach leverages routinely collected administrative data, starting from a large baseline cohort of individuals from the population of interest and linked events, to estimate the full size of the target population. When multiple event types are included, the approach enables checking of the consistency of evidence about prevalence from different event types. Additional evidence can be incorporated where inconsistencies are identified. In this article, we summarize the general framework of MPEP, with focus on the most recent version, with improved computational efficiency (implemented in STAN). We also explore several extensions to the model that help us understand the sensitivity of the results to modelling assumptions or identify potential sources of bias. We demonstrate the MPEP approach through a case study estimating the prevalence of opioid dependence in Scotland each year from 2014 to 2022.
Sam: Yes. They track person-time—the total years people in each subgroup are followed—separately for on-treatment and off-treatment periods, because rates differ a lot. Events must be specific to opioid users, like opioid-related overdoses or admissions, so they don't count non-users and inflate the extra count. In the Scotland case, they defined on-treatment using prescription reimbursement dates, starting 60 days before and ending 12 days before, merging close episodes to capture real patterns.
Alex: That specificity for events sounds crucial—otherwise, you'd mix in unrelated cases.
Sam: Precisely, or you'd overestimate. A big edge over simple multipliers is adjusting event rates for factors like treatment status or demographics, making estimates more consistent across event types. The paper notes this setup informs policy for harm reduction, though they stress it's illustrative.
Alex: But to make those adjustments work, they must have a way to count the time those hidden people are at risk for events, right?
Sam: Correct. For events not linked to the baseline cohort, they model the count so the expected number matches what you'd see if the risk level was the same as off-treatment in the baseline group. This uses a counting process that tracks rare events over time, like logging fish bites in a lake based on known bait spots.
Alex: So the hidden group's event count gets explained by borrowing the off-treatment risk from the seen group... that anchors the scaling.
Sam: Yes. They also calculate the total time the extra population is exposed to risk—person-years at risk—separately for those who died from opioid events and the rest. To avoid assuming everyone survives the full year, they adjust using restricted mean survival time: the average fraction of the year people in that group would live, assuming steady background exit risks from off-treatment baseline data. This makes the exposure realistic.
Alex: Huh, so no overcounting exposure... Okay, and how do they tie it all to prevalence?
Sam: They split total prevalence into observed and unobserved parts, then fit a separate math model just to the unobserved share—the proportion of the general population in the extra group. This boosts stability, letting runs finish in minutes and handling finer splits like more age bands or regions. In the Scotland case, using opioid deaths and hospital admissions, they stratified by sex, three age groups from 15 to 64, years 2014-15 to 2022-23, and eight regions.
Alex: That separate modeling for the hidden part sounds like it avoids computational snags... but they pick covariates carefully for those rate models?
Sam: Exactly. Each rate model starts with main factors like age, sex, year, and treatment status, plus targeted interactions from prior knowledge. They select the best fit using stats like deviance information criterion, add random effects for region-year combos, and implement in Stan software via R—code's on GitHub. Multiple events let joint checks for consistency.
Alex: Those random effects and model selection sound solid... but what if the count data has quirks like too many zeros or extra variation?
Sam: Good point. Sometimes event counts have more zeros than expected, like in small regions with no events. Or the spread of counts is wider than predicted, called overdispersion, because real risks vary due to hidden factors. For extra zeros, they suggest a zero-inflated Poisson model, which splits data into a group that never has events and one that follows normal counts. For overdispersion, a negative binomial model adds a mixing factor, like blending counts from slightly different risk levels.
Alex: So it's like adjusting for empty buckets or wobbly averages... makes sense for sparse opioid events.
Sam: In the Scotland application, they tested these on opioid deaths and admissions. Prevalence estimates stayed largely consistent between plain Poisson, zero-inflated, and negative binomial versions, though intervals widened a bit with the fancier ones.
Alex: Okay, consistent estimates across tweaks is reassuring... and with multiple events, do they check if deaths and admissions agree?
Sam: Yes, using node-splitting: they compare prevalence from deaths alone versus admissions alone, then see if joint use fits without conflict. A consistency p-value near 0.5 means good agreement; here most years hovered around 0.2 to 0.7, suggesting reliable evidence.
Alex: Huh, so the adjustments keep things robust... without overpromising precision.
Alex: But those consistency checks aren't always perfect—what happens when event types point to different totals?
Sam: When node-splitting reveals a conflict—like a consistency p-value under 0.05—they explore why. One approach is adding a bias parameter to the event rate model for the suspect source. This multiplier tweaks the assumed off-treatment risk level to test how much violation of the equal-rate assumption explains the clash. They give it a non-informative prior centered at no bias with wide uncertainty.
Alex: So it's like calibrating one scale against another when they don't match.
Sam: Exactly. In the Scotland data for 2022/23, applying this to admissions pulls the joint prevalence estimate closer to the deaths-only figure, though gaps remain. The bias terms vary notably across subgroups, highlighting uncertainty.
Alex: Huh, and for other data quirks, like imperfect linking between records?
Sam: Yes, they introduce a linkage probability parameter, pmatch—the chance a true event in the baseline cohort gets correctly matched. Below 1 means some cohort events appear as "extra," underestimating cohort rates and inflating hidden sizes. The model adjusts by scaling cohort event expectations down by pmatch and adding back unlinked ones, using an informative prior from external evidence.
Alex: So these tweaks make the method flexible for real-world messiness... without claiming perfection.
Sam: Precisely—the paper stresses they're tools for diagnosis, not fixes; they preserve caution, ensuring estimates inform policy like harm reduction while noting limitations.
Alex: So pulling it all together, this method gives policymakers in Scotland a clearer picture of opioid dependence numbers year by year... more reliably than older approaches.
Sam: Yes, by linking multiple event types and adjusting for real-world issues like mismatches or extra variation. The case study from 2014 to 2022 shows how it supports harm reduction planning.
Alex: Right, but it relies on some key setup pieces—like a baseline cohort and specific events.
Sam: That's right. It needs events highly specific to opioid users so unrelated cases don't creep in. Assumptions like equal off-treatment event rates between seen and hidden groups must hold, and everything hinges on good linked data. Without at least two event types, you lose those consistency checks.
Alex: And for other hidden groups, like cocaine users, those treatment patterns might differ a lot.
Sam: Exactly—for stimulants without standard treatment, fewer people might be in records. Still, as admin data grows worldwide, the approach adapts well for better interventions.
Alex: Huh, so it's a step toward routine counts from everyday records.
Sam: Precisely—a flexible tool that turns linked data into actionable prevalence estimates, with built-in ways to spot and handle flaws. The paper positions it as illustrative for opioid work in Scotland, but flags the need to validate assumptions per setting. Overall, it offers a meaningful way forward for sizing up stigmatized populations to guide public health.
Alex: That's a solid overview, Sam—thanks for breaking it down so clearly. This has been a useful look at turning messy records into reliable numbers for real needs. Thanks for listening to ResearchPod.