Depletion of Susceptibles

Depletion of susceptibles—also known in biostatistics, causal inference, and pharmacoepidemiology as differential frailty selection, attrition of susceptibles, or the "built-in selection bias of hazard ratios"—is a time-varying selection distortion in longitudinal cohort studies and randomized clinical trials wherein individuals who possess a high baseline genetic, immunological, or physiological vulnerability ("frailty") to an outcome experience the event rapidly after exposure initiation and are removed from the at-risk denominator, progressively enriching the exposed survivor pool with low-risk ("immune" or resilient) individuals. First analyzed mathematically in demographic frailty models by James W. Vaupel, Kenneth G. Manton, and Eric Stallard (1979), clinically characterized in pharmacoepidemiology by Yola Moride and Lucien Abenhaim (1994), and formalized within modern counterfactual causal diagrams by Miguel A. Hernán (2010) and Odd O. Aalen (2015), depletion of susceptibles explains why the period-specific hazard ratio (HR(t)) for a genuinely harmful exposure (such as postmenopausal hormone therapy, NSAIDs, or smoking) routinely attenuates toward 1.0—and can even invert below 1.0 (making a harmful exposure appear artificially protective!)—in later follow-up intervals, even inside a perfectly randomized clinical trial with zero loss to follow-up and zero baseline confounding.

Why the Hazard Ratio Conditions on a Collider (S_t = 1)

Suppose an investigator conducts an ideal randomized controlled trial comparing an active drug (Exposure E = 1) against placebo (Exposure E = 0), so that at baseline (t = 0) treatment E is statistically independent of all unmeasured genetic or physiological susceptibility factors (Frailty U, such as Factor V Leiden thrombophilia or CYP2C9 polymorphism). Suppose further that the drug never protects anyone—it only increases (or leaves unchanged) the instantaneous risk of the outcome (such as venous thromboembolism or myocardial infarction) during Period 1 (t ∈ [0, 1]) and Period 2 (t ∈ [1, 2]) by interacting with the unmeasured frailty U.

Why does a bias emerge in Period 2? Because the hazard rate at time t is defined as the instantaneous probability of experiencing the event at time t conditional on having survived event-free up to time t (D₁ = 0):

h(t | E) = lim_Δt→0 P(T ∈ [t, t + Δt) | T ≥ t, E) / Δt

In the language of causal directed acyclic graphs (DAGs), survival past Period 1 (D₁ = 0) is a common effect (a collider) of both Treatment (E) and Unmeasured Frailty (U):

Treatment (E) → Early Event / Survival (D₁ = 0) ← Unmeasured Susceptibility (U) → Late Event (D₂)

Because the Period-2 hazard ratio HR₂ = P(D₂ = 1 | D₁ = 0, E = 1) / P(D₂ = 1 | D₁ = 0, E = 0) restricts the denominator to individuals who did not fail in Period 1 (D₁ = 0), calculating a period-specific hazard ratio conditions on the collider D₁ = 0! This opens the non-causal backdoor path E → [D₁ = 0] ← U → D₂:

• In the Exposed arm (E = 1), the combination of E = 1 and high frailty U = 1 caused the most susceptible individuals to suffer their event early during Period 1 and exit the risk set. Thus, by the start of Period 2, the surviving E = 1 risk set has been depleted of susceptibles (U = 1).
• In the Unexposed control arm (E = 0), many susceptible individuals (U = 1) survived Period 1 because they were not exposed to the drug trigger, so they remain inside the E = 0 risk set entering Period 2!
• Consequently, in Period 2, an artificially resilient survivor pool in the exposed arm (E = 1) is being compared against a frailer survivor pool in the unexposed arm (E = 0), driving the Period-2 hazard ratio HR₂ downward—often below 1.0!

Numerical Proof of Hazard Ratio Inversion (HR₁ = 2.60 → HR₂ = 0.69)

Consider a randomized trial with 10,000 patients in the Exposed arm (E = 1) and 10,000 patients in the Placebo arm (E = 0). In each arm at baseline (t = 0), 20% of patients (n = 2,000) are genetically Susceptible (U = 1) and 80% (n = 8,000) are Resilient (U = 0). Suppose that in every single time period, the drug doubles event risk in Susceptible patients (from 25% per period on placebo to 50% per period on drug) and increases risk by 50% in Resilient patients (from 1.0% per period on placebo to 1.5% per period on drug). Notice that the drug is harmful to every single person in every single period (RR_U=1 = 2.00 and RR_U=0 = 1.50 in both Period 1 and Period 2):

If we continue this exact table into Period 4 (t = 3 to 4 yr), only 250 susceptibles remain in the Exposed arm (producing 125 + 113 = 238 events among 7,774 survivors = 3.06%), whereas 844 susceptibles still remain in the Placebo arm (producing 211 + 77 = 288 events among 8,530 survivors = 3.38%), yielding a Period-4 Hazard Ratio of HR₄ = 3.06% / 3.38% = 0.90 < 1.0—making a purely harmful drug look statistically protective after year 3 solely because it killed or injured its susceptible takers earlier!

1. The Women's Health Initiative (WHI) Estrogen-Plus-Progestin Trial (Hernán, 2010)

In Miguel A. Hernán's 2010 landmark paper "The Hazards of Hazard Ratios" in Epidemiology, Hernán examined the randomized Women's Health Initiative (WHI) trial comparing oral conjugated equine estrogens plus medroxyprogesterone acetate against placebo for coronary heart disease (CHD) in 16,608 postmenopausal women:

• Year 1: HR = 1.81 (81% increased hazard of acute CHD events as prothrombotic estrogen precipitated plaque rupture/thrombosis in women with subclinical prothrombotic/atherosclerotic susceptibility).
• Year 2 to Year 5: HR = 1.34 → 1.27 → 1.25 → 1.45 (progressive attenuation as high-risk women exited the risk set).
• Year 6+: HR = 0.70 (95% CI: 0.42–1.14) — an apparent 30% protective reduction in Year 6+ that misled some commentators into claiming hormone therapy becomes "cardioprotective after 5 years of use," when in reality cumulative CHD survival curves never crossed; the exposed arm simply ran out of prothrombotic susceptibles by Year 6!

2. Prevalent-User vs. New-User Designs in Pharmacoepidemiology (NSAIDs and GI Bleeding)

In 1994, Yola Moride and Lucien Abenhaim demonstrated depletion of susceptibles in observational studies of nonsteroidal anti-inflammatory drugs (NSAIDs) and upper gastrointestinal bleeding. Patients with underlying Helicobacter pylori colonization or defective mucosal prostaglandin synthesis develop peptic ulcer bleeding within the first 30 to 90 days of starting an NSAID and immediately discontinue the drug.

• Consequently, prevalent NSAID users (patients who have already tolerated an NSAID for >1 year without bleeding) represent a heavily selected survivor cohort depleted of susceptibles (analogous to the healthy worker effect and incidence-prevalence bias).
• As formalized by Wayne A. Ray (2003), enrolling prevalent users into an observational cohort mixes depleted long-term survivors with new initiators, severely underestimating true drug toxicity (and explain why observational hormone therapy cohorts before WHI—which enrolled prevalent long-term estrogen users who had already survived the early Year-1 thrombotic window—falsely reported an overall HR ≈ 0.60 for coronary heart disease).

3. Age-Related Attenuation of Smoking, Obesity, and APOE ε4 Hazard Ratios

In geriatric epidemiology, the hazard ratios for cigarette smoking, hypertension, hypercholesterolemia, and the APOE ε4 allele (for Alzheimer's disease) consistently decline at advanced ages (e.g., age 85–95 compared with age 55–65) and occasionally drop below 1.0 among nonagenarians. Rather than proving that smoking or APOE ε4 becomes biologically benign at age 90, this crossover reflects differential frailty depletion: smokers and APOE ε4 carriers who lacked compensatory longevity alleles died or developed disease in their 60s and 70s, leaving only exceptionally resilient survivors in the 90-year-old exposed risk set.

To prevent depletion of susceptibles from distorting causal inference, epidemiologists and biostatisticians implement three core remedies:

1. The Active-Comparator New-User (ACNU) Design (Ray, 2003; Lund et al., 2015): Never enroll prevalent drug users. Always anchor time zero (t = 0) at the exact first prescription after a washout period (new users) so that the early high-susceptibility window (Months 0–6) is fully captured prior to susceptible depletion, while avoiding immortal time bias.
2. Report Standardized Cumulative Survival / Risk Curves (S(t) and F(t) = 1 − S(t)) Rather Than Period-Specific Hazard Ratios (Hernán, 2010): Unlike the hazard ratio h(t | E = 1) / h(t | E = 0)—which conditions on the post-baseline collider T ≥ t—the marginal cumulative incidence (risk) difference F(t | E = 1) − F(t | E = 0), risk ratio F(t | E = 1) / F(t | E = 0), and restricted mean survival time (RMST) difference compare the entire baseline randomized cohorts (N₁ vs. N₀) without ever conditioning on post-randomization survival! In our numerical table above, cumulative risk by the end of Period 4 is 22.1% in the Exposed arm versus 15.8% in the Placebo arm (RR_cum = 1.40 > 1.0)—remaining unambiguously harmful at every time point t.
3. Shared Frailty and Accelerated Failure Time (AFT) Models (Aalen et al., 2015): Explicitly model the unobserved baseline frailty distribution U ~ Gamma(1, θ) or use accelerated failure time (AFT) models, which remain collapsible across unmeasured baseline covariates.

Why did Miguel Hernán call the hazard ratio "a measure with built-in selection bias"?

Even in a randomized clinical trial with 100% adherence and zero loss to follow-up, randomization only guarantees that the exposed (E = 1) and unexposed (E = 0) groups are exchangeable (balanced on unmeasured susceptibility factors U) at baseline (t = 0). As soon as events begin occurring after t > 0, if the treatment has any causal effect on the event, the treatment preferentially removes susceptible individuals (U = 1) from one arm faster than the other. Because the hazard at time t is calculated only among survivors (T ≥ t), the two survivor groups at t > 0 are no longer exchangeable on U. Conditioning on survival T ≥ t is conditioning on a post-treatment collider, which builds selection bias directly into the mathematical definition of any period-specific hazard ratio.

How did depletion of susceptibles cause observational studies in the 1980s and 1990s to conclude that postmenopausal hormone therapy prevented heart attacks?

In the randomized WHI trial, oral hormone replacement therapy increased coronary heart disease events by 81% during Year 1 (HR = 1.81) as women susceptible to thrombosis suffered early myocardial infarctions, and then attenuated to HR ≤ 1.0 in later years once those susceptibles were depleted. Observational cohort studies in the 1980s and 1990s (such as the original Nurses' Health Study analyses) enrolled prevalent hormone users—women who had already been taking estrogen for years at baseline and had survived and passed through the dangerous Year-1 thrombotic window prior to study entry. By missing the early spike in susceptible women and observing only the depleted long-term survivor phase (combined with healthy-user confounding), prevalent-user cohorts reported a spurious protective effect.

Does a declining hazard ratio over time always mean that a vaccine is waning or a drug's toxicity is wearing off?

No! Even if a drug's biological toxicity (or a vaccine's true biological protection) remains 100% constant every single day for every individual, heterogeneity in baseline frailty or exposure intensity across the population mathematically forces the observed population hazard ratio HR(t) to converge toward 1.0 over follow-up time as high-risk susceptibles in the unvaccinated/exposed group are infected or injured early and exit the risk set. Distinguishing true biological waning of vaccine efficacy from depletion of susceptibles in the unvaccinated control arm requires staggered-enrollment comparisons or frailty-adjusted estimators.

1. Hernán, M. A. (2010). "The hazards of hazard ratios." Epidemiology, 21(1), 13–15. https://doi.org/10.1097/EDE.0b013e3181c1ea43
2. Vaupel, J. W., Manton, K. G., & Stallard, E. (1979). "The impact of heterogeneity in individual frailty on the dynamics of mortality." Demography, 16(3), 439–454. https://doi.org/10.2307/2061224
3. Moride, Y., & Abenhaim, L. (1994). "Evidence of the depletion of susceptibles effect in non-experimental pharmacoepidemiologic research." Journal of Clinical Epidemiology, 47(7), 731–737. https://doi.org/10.1016/0895-4356(94)90170-890170-8)
4. Ray, W. A. (2003). "Evaluating medication effects outside of clinical trials: new-user designs." American Journal of Epidemiology, 158(9), 915–920. https://doi.org/10.1093/aje/kwg231