Prior Event Rate Ratio (PERR) Analysis

• PERR is a methodological framework that uses pre-treatment event rates to cancel out time-constant confounding in observational studies.
• It partitions follow-up into prior and post periods, enabling estimation of treatment effects via ratios of hazard ratios or risk ratios.
• Extensions, including Andersen–Gill and conditional frailty models, address challenges like unobserved heterogeneity, dropout, and event dependence.

Searching arXiv for recent and foundational papers on PERR to ground the article in the provided literature. Prior Event Rate Ratio (PERR) is a design–analysis strategy for observational studies that uses pre-treatment outcome experience to adjust for measured and unmeasured confounding, especially when confounding is time-constant across observation periods. The method partitions follow-up into a prior period, in which no one has yet received treatment, and a post period, in which the treated group has initiated treatment, and then estimates treatment effect by taking the ratio of the treated-versus-untreated association in the post period to the corresponding association in the prior period. In time-to-event settings this is a ratio of hazard ratios; for binary outcomes it can be expressed as a ratio of risk ratios. Recent work has emphasized that PERR is better understood as a methodological framework rather than a single estimator, because its operating characteristics depend materially on how periods, outcomes, recurrent events, and dropout are handled (Cheung et al., 27 May 2025, Ma et al., 2024).

1. Core construction and estimands

The canonical PERR setup distinguishes an “ever treated” group, denoted $X=1$, from a “never treated” group, denoted $X=0$. For binary outcomes, $Y_1 \in \{0,1\}$ denotes the prior-period event and $Y_2 \in \{0,1\}$ the post-period event. The central intuition is that, in the prior period, any difference in event risk between $X=1$ and $X=0$ cannot be due to treatment because treatment has not yet started; it therefore reflects confounding. In the post period, the treated-versus-untreated difference reflects both confounding and treatment effect. Under time-constant confounding, the confounding component is the same in both periods and cancels in the ratio (Cheung et al., 27 May 2025).

For binary outcomes, the generic PERR estimand is

$$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$

In the time-to-event formulation used in the classical literature, two separate Cox models yield $HR_{\text{prior}}$ and $HR_{\text{post}}$, and the PERR estimate is

$$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$

A single-model reformulation writes the hazard as

$X=0$0

with $X=0$1 for those who ever initiate treatment and $X=0$2 after the treatment or index date. In that parameterization,

$X=0$3

This interaction-based expression is algebraically equivalent to the two-model PERR but yields the PERR hazard ratio directly as the treatment-by-period interaction (Ma et al., 2024).

2. Identifying assumptions and interpretive scope

PERR relies first on a time-constant confounding assumption: the influence of confounders, including unmeasured confounders, on the outcome must be stable across the prior and post periods. In the multiplicative hazard formulation, if confounding inflates the treated-versus-untreated comparison by the same factor in both periods, then that factor cancels in the ratio. If the magnitude of confounding changes between periods, the cancellation fails and PERR is biased (Ma et al., 2024).

Several further assumptions have been isolated in recent work. One concerns population homogeneity or, conversely, the presence of unobserved heterogeneity or frailty. Standard first-event Cox implementations of PERR can be biased when frailty is non-zero, because high-risk individuals fail early and leave the risk set, thereby altering the composition of those remaining at risk over time. Another concerns event-independent treatment, also called absence of event-dependent treatment (EDT): prior outcome events should not affect the probability or timing of treatment initiation. If treatment is triggered or accelerated by recent events, the prior-period event rate among future-treated individuals can be artificially elevated, which distorts $X=0$4 and biases the PERR ratio (Ma et al., 2024).

A distinct assumption, emphasized in recurrent-event settings, is absence of event dependence: previous outcome events should not themselves alter the hazard of later events beyond measured confounders and treatment. Violation of that condition can also bias crude PERR estimates, because treated and untreated groups may differ systematically in prior event counts, and those prior events then modify future risk. The provided literature also notes non-differential mortality as a relevant assumption, or alternatively the need to adjust mortality-related bias by joint modelling; this becomes especially salient when post-period outcomes are observed only among completers (Ma et al., 16 Jul 2025, Cheung et al., 27 May 2025).

These assumptions delimit the interpretive scope of PERR. The method is designed for settings with a well-defined pre-treatment window, a post-initiation window, and confounding that is plausibly stable over time. This suggests that PERR is most naturally interpreted as a longitudinal confounding-control strategy rather than a universal substitute for outcome regression.

3. Estimator variants and model extensions

Recent papers distinguish several PERR implementations that differ in their target population and in the event process being modeled. For binary outcomes with post-period mortality or dropout, two estimators have been contrasted explicitly: $X=0$5 and

$X=0$6

Here $X=0$7 indicates death or dropout in the post period and $X=0$8 indicates a completer. $X=0$9 combines a post-period ratio among completers with a prior-period ratio in the full population, whereas $Y_1 \in \{0,1\}$0 uses completers in both periods and therefore compares like with like within the same selected subpopulation (Cheung et al., 27 May 2025).

For recurrent-event data, two major extensions have been proposed. The Andersen–Gill implementation, denoted $Y_1 \in \{0,1\}$1, embeds the treatment-by-period interaction in a counting-process model using all recurrent events and robust standard errors for clustering. Simulations reported in the literature show that $Y_1 \in \{0,1\}$2 remains approximately unbiased under unobserved heterogeneity in situations where first-event Cox implementations show growing bias and poor coverage (Ma et al., 2024).

When recurrent events are themselves event-dependent, a conditional frailty extension has been proposed: $Y_1 \in \{0,1\}$3 where $Y_1 \in \{0,1\}$4 indexes event order, $Y_1 \in \{0,1\}$5 is an event-stratum-specific baseline hazard, and $Y_1 \in \{0,1\}$6 is a person-level frailty. In this specification, $Y_1 \in \{0,1\}$7 remains the PERR hazard ratio, now estimated after conditioning on event number and allowing subject-specific heterogeneity (Ma et al., 16 Jul 2025).

Estimator or model	Core comparison	Principal issue addressed
$Y_1 \in \{0,1\}$8	Prior full population; post completers	Baseline PERR with dropout mismatch
$Y_1 \in \{0,1\}$9	Prior and post among completers	Differential mortality/dropout
$Y_2 \in \{0,1\}$0	Recurrent-event AG interaction model	Unobserved heterogeneity in first-event Cox PERR
$Y_2 \in \{0,1\}$1	Event-stratified frailty PERR	Event dependence in recurrent outcomes

The proliferation of these variants underlies the statement that “The PERR is better seen as a methodological framework. Its performance depends on the specifications within the framework” (Cheung et al., 27 May 2025).

4. Differential mortality or dropout

A 2025 simulation study by Cheung and Ma examined PERR performance when post-period outcome $Y_2 \in \{0,1\}$2 is observed only among those with $Y_2 \in \{0,1\}$3, and replicated the earlier Uddin et al. scenario in which mortality or dropout is simultaneously influenced by treatment, confounder, and prior event. The simulations used $Y_2 \in \{0,1\}$4 per dataset, 10,000 replicates per scenario, and post-period mortality or dropout rates of 0\%, 5\%, 10\%, 15\%, and 20\%. The true treatment effect on $Y_2 \in \{0,1\}$5 was fixed at a risk ratio of $Y_2 \in \{0,1\}$6 (Cheung et al., 27 May 2025).

Four scenarios varied the determinants of $Y_2 \in \{0,1\}$7: $Y_2 \in \{0,1\}$8; $Y_2 \in \{0,1\}$9; $X=1$0; and $X=1$1 alone. In the original scenario, where $X=1$2 depends on confounder, prior event, and treatment, $X=1$3 showed underestimation bias that worsened as mortality or dropout increased; at 20\% mortality or dropout its mean estimate was $X=1$4 for a true risk ratio of $X=1$5. In the same scenario, $X=1$6 was only mildly biased upward; at 20\% mortality or dropout its mean estimate was $X=1$7, and its absolute bias was about one-third that of $X=1$8. In the scenario where dropout depends on $X=1$9 and $X=0$0 but not $X=0$1, $X=0$2 remained biased, with mean approximately $X=0$3 at 20\% dropout, whereas $X=0$4 was nearly unbiased at approximately $X=0$5. The crude post-period relative risk among completers remained strongly confounded, reaching approximately $X=0$6 at 20\% dropout (Cheung et al., 27 May 2025).

The decisive structural distinction was whether dropout depended on prior event status. In scenarios where $X=0$7 did not depend on $X=0$8, namely the $X=0$9 and $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$0-only scenarios, $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$1 provided accurate estimates across all simulated dropout levels, even when dropout was treatment-dependent. The proposed explanation is a selection-bias or collider-stratification mechanism: conditioning on $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$2 is particularly problematic when $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$3 is directly affected by $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$4, because $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$5 also appears in the PERR denominator. By contrast, $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$6 is generally more vulnerable because it compares a selected post-period population with an unselected prior-period population. The resulting conclusion was that $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$7 provides unbiased estimates unless mortality or dropout is affected by prior event (Cheung et al., 27 May 2025).

5. Event-dependent treatment and recurrent-event reformulation

A separate line of work has addressed event-dependent treatment, meaning that prior outcome events affect the hazard of treatment initiation. In the formulation proposed for recurrent events, if $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$8 is the $$\text{PERR} = \frac{\text{Post-period risk ratio treated vs untreated}} {\text{Prior-period risk ratio treated vs untreated}}.$$9-th event time for person $HR_{\text{prior}}$0, EDT is represented by

$HR_{\text{prior}}$1

where $HR_{\text{prior}}$2 is the duration of event influence and $HR_{\text{prior}}$3 its multiplicative effect on treatment hazard. When $HR_{\text{prior}}$4, events increase treatment hazard, so future-treated individuals tend to accumulate excess events immediately before treatment initiation; this inflates $HR_{\text{prior}}$5 and biases PERR downward. When $HR_{\text{prior}}$6, the opposite direction is expected (Ma et al., 2024).

To address frailty-induced bias and EDT simultaneously, PERR has been reformulated in an Andersen–Gill recurrent-event framework, $HR_{\text{prior}}$7, and supplemented by an EDT detection strategy based on “gap” intervals in the prior period. The prior period $HR_{\text{prior}}$8 is partitioned into a far segment and $HR_{\text{prior}}$9 contiguous sub-periods of width $HR_{\text{post}}$0 immediately preceding treatment or index time. An AG model with treatment, gap indicators, and treatment-by-gap interactions is fit on prior-period data only; statistically significant terminal-gap interactions are interpreted as evidence of EDT, and consecutive significant terminal gaps define an estimated EDT duration $HR_{\text{post}}$1. A simplified model then estimates $HR_{\text{post}}$2, after which control-group index times are algorithmically shifted to induce analogous event dependence in controls. The corrected treatment effect is then obtained by refitting $HR_{\text{post}}$3 to treated subjects and EDT-adjusted controls (Ma et al., 2024).

Simulation evidence reported for this strategy showed that original PERR and the single-Cox interaction formulation are essentially identical, that both are systematically biased when frailty variance is non-zero, and that $HR_{\text{post}}$4 remains approximately unbiased with appropriate coverage and smaller RMSE in those settings. Further simulations with EDT showed that uncorrected $HR_{\text{post}}$5 is biased in the predicted direction, whereas the correction method yields mean estimates close to the true hazard ratio and coverage probabilities much closer to 95\% across a range of scenarios (Ma et al., 2024).

6. Event dependence and the conditional frailty approach

Event dependence is distinct from EDT. Here the concern is that occurrence of previous outcome events alters the hazard of later events. In the recurrent-event simulations used to study this issue, the outcome hazard was modeled as

$HR_{\text{post}}$6

with $HR_{\text{post}}$7 a gamma frailty, $HR_{\text{post}}$8 an unobserved binary confounder, $HR_{\text{post}}$9 an observed binary covariate, and $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$0 an event-dependence term. Two forms were considered: constant event dependence,

$$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$1

and transient event dependence with decaying effect since the last event. Negative event dependence means prior events reduce later event rates; positive event dependence means they increase later event rates (Ma et al., 16 Jul 2025).

Under these departures from the standard PERR assumption of no event dependence, crude $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$2 was biased systematically. In the reported simulations, negative event dependence biased crude PERR toward the null, while positive event dependence biased it away from the null, sometimes severely. For example, with positive constant event dependence and a true hazard ratio of $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$3, relative bias for $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$4 reached $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$5; with negative transient event dependence and a true hazard ratio of $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$6, relative bias was $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$7 and coverage probability fell to $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$8. The proposed conditional frailty implementation, $$\text{PERR HR}=\frac{HR_{\text{post}}}{HR_{\text{prior}}}.$$9, reduced the median absolute relative bias to about $X=0$00, decreased RMSE, and generally improved coverage toward 95\% (Ma et al., 16 Jul 2025).

The same paper illustrated the method using the COMPASS cohort of 600 adults with stage IV solid cancers in Singapore, reduced to 575 after exclusions. Community-based palliative care recipients were matched 1:1 to non-recipients, producing a PERR cohort of 370 patients and a prior/post window restricted to 70 days before and after treatment or index time. Dynamic Random-Intercept Modeling found positive event dependence in emergency department visits, with odds ratio $X=0$01 (95\% CI $X=0$02–$X=0$03; $X=0$04). In the full cohort, conventional Andersen–Gill regression with covariate adjustment estimated a hazard ratio of $X=0$05 (95\% CI $X=0$06–$X=0$07) for palliative care versus no palliative care. In the matched PERR cohort, crude $X=0$08 yielded $X=0$09 (95\% CI $X=0$10–$X=0$11), whereas $X=0$12 yielded $X=0$13 (95\% CI $X=0$14–$X=0$15), moving the estimate further toward the null in a direction consistent with correction for positive event dependence (Ma et al., 16 Jul 2025).

7. Applications, related designs, and current methodological position

PERR has been used in real-world evidence research and pharmacoepidemiology, and papers place it alongside other longitudinal strategies that exploit within-person or pre-period information. One line of comparison connects PERR with self-controlled case series and dynamic random intercept models; another notes its conceptual similarity to difference-in-differences, in that both rely on a stable-structure assumption across periods. The distinction is that PERR estimates a ratio of treated-versus-untreated associations across periods rather than a difference in group-level changes. The literature summarized here also describes PERR as a between-group analogue of self-controlled approaches, using pre-period data to quantify and adjust for stable confounding in the post period (Cheung et al., 27 May 2025).

Applied illustrations in advanced cancer show how sensitive substantive conclusions can be to the particular PERR specification and to assumption checking. In one palliative care study using National Cancer Centre Singapore data, naive post-period Cox analysis suggested that patients who had started receiving palliative care had higher incidence of emergency department visits than their matched controls, with hazard ratio $X=0$16 (95\% CI $X=0$17 to $X=0$18). Standard PERR based on Cox models suggested a $X=0$19 reduction, $X=0$20 (95\% CI $X=0$21 to $X=0$22), while the recurrent-event $X=0$23 estimate was $X=0$24 (95\% CI $X=0$25 to $X=0$26). After detection of EDT in the five weeks before palliative-care initiation and correction using $X=0$27 weeks and $X=0$28, the corrected $X=0$29 estimate became $X=0$30 (95\% CI $X=0$31 to $X=0$32) (Ma et al., 2024). This sequence does not imply that PERR is unreliable as such; rather, it indicates that distinct assumption violations can move estimates in different directions and that specification matters materially.

The current methodological position in the provided literature is therefore differentiated rather than uniformly skeptical. Under differential mortality or dropout, a completer-only binary-outcome implementation can be unbiased unless dropout is affected by prior event. Under unobserved heterogeneity in recurrent-event settings, the Andersen–Gill reformulation improves robustness relative to first-event Cox PERR. Under event dependence, conditional frailty substantially reduces bias, although the same paper cautions that applying conditional frailty when event dependence is absent may introduce modest bias and slightly reduced coverage. Open questions identified in these papers include the performance of alternative PERR implementations for event-rate outcomes under informative censoring and the broader challenge of jointly handling time-constant confounding, event dependence, and differential mortality within one coherent framework (Cheung et al., 27 May 2025, Ma et al., 16 Jul 2025).