Moderated Causal Excursion Odds Ratio

• Moderated causal excursion odds ratio is a causal estimand that quantifies how the effect of a time-varying intervention on binary outcomes varies with prespecified moderators in micro-randomized trials.
• It employs an odds‐ratio scale alongside best linear projection methods to extract low-dimensional targets while conditioning on key summary variables of treatment history.
• The approach supports robust estimation in both individual-level and clustered settings, addressing between-cluster heterogeneity and within-cluster interference in mobile health studies.

Moderated causal excursion odds ratio is a causal estimand for micro-randomized trials (MRTs) with longitudinal binary outcomes that quantifies how the effect of a time-varying intervention varies with prespecified moderators at a given decision time. It is the odds-ratio-scale analogue of the causal excursion effect developed for mobile health and related sequentially randomized settings, and it is used to characterize marginal or moderated intervention effects while conditioning only on chosen summary variables rather than the full treatment and covariate history. In the recent literature, the construct appears both in standard individual-level MRTs and in clustered settings with possible between-cluster treatment effect heterogeneity and within-cluster interference (Yu et al., 24 Sep 2025).

1. Conceptual definition

For a generic participant at decision point $t=1,\dots,T$, let $A_t\in\{0,1\}$ denote randomized treatment, $Y_{t+1}\in\{0,1\}$ a binary proximal outcome, $X_t$ the prespecified moderator(s), $H_t$ the history up to just before $A_t$, and $I_t\in\{0,1\}$ the availability indicator. Under standard consistency, positivity, and sequential ignorability assumptions, the moderated causal excursion odds ratio at time $t$ for moderator value $x$ is defined by

$$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$

This quantity compares the odds of the proximal binary outcome under treatment versus control at the same decision time, within a stratum of the moderator. To obtain a low-dimensional target, the literature defines a best linear projection

$A_t\in\{0,1\}$0

where $A_t\in\{0,1\}$1 is a user-chosen feature vector and $A_t\in\{0,1\}$2 are weights with $A_t\in\{0,1\}$3 (Yu et al., 24 Sep 2025).

The odds-ratio formulation is distinct from the earlier relative-risk formulation, but it addresses the same scientific objective: estimation of a time-varying moderated intervention effect in an MRT. A practical implication is that the target is explicitly scale-specific. This is consistent with the recommendation that causal risk difference, relative risk, and odds ratio all be reported in MRTs with binary outcomes, since effect moderation can differ in direction across scales (Yu et al., 24 Sep 2025).

2. Origins in causal excursion effects for binary outcomes

The immediate precursor is the moderated causal excursion effect for binary proximal outcomes on the log-relative-risk scale. In that formulation, with $A_t\in\{0,1\}$4 denoting a moderator formed from $A_t\in\{0,1\}$5, the causal excursion effect is

$A_t\in\{0,1\}$6

often modeled parametrically as $A_t\in\{0,1\}$7 (Qian et al., 2019).

That framework was developed for MRTs in which each individual is repeatedly randomized among intervention options over many decision times and the primary outcome may be a longitudinal binary outcome. The estimation strategy introduced there used a working nuisance model for the counterfactual mean under no treatment,

$A_t\in\{0,1\}$8

a “blipped-down” outcome $A_t\in\{0,1\}$9, and an inverse-probability-weight coupling $Y_{t+1}\in\{0,1\}$0 that both holds subsequent treatments at $Y_{t+1}\in\{0,1\}$1 and recenters treatment assignment from the full history $Y_{t+1}\in\{0,1\}$2 to the lower-dimensional moderator $Y_{t+1}\in\{0,1\}$3 (Qian et al., 2019).

Within that literature, the odds-ratio-scale version was not formally developed in full detail. The explicit statement was that one could in principle replace the log-link by a logit-link to target odds ratios, replace the blipped-down factor by the appropriate logistic structural nested mean model g-estimator weight, and carry the semiparametric efficiency machinery through with the redefined efficient score. This places moderated causal excursion odds ratio as a direct methodological continuation rather than a separate conceptual object (Qian et al., 2019).

3. Direct and indirect log-odds excursion effects under clustering and interference

A more general formulation arises when MRT data are clustered and the binary outcome may be affected not only by an individual’s own treatment but also by treatment assigned to others in the same cluster. Let clusters $Y_{t+1}\in\{0,1\}$4 have size $Y_{t+1}\in\{0,1\}$5. For individual $Y_{t+1}\in\{0,1\}$6 in cluster $Y_{t+1}\in\{0,1\}$7 at decision time $Y_{t+1}\in\{0,1\}$8, let $Y_{t+1}\in\{0,1\}$9 be the collection of past treatments $X_t$0 and covariates $X_t$1, and let $X_t$2 be randomized with known probability $X_t$3. Let $X_t$4 denote the binary outcome measured $X_t$5 steps after $X_t$6, with potential outcome

$X_t$7

Let $X_t$8 be a vector of effect moderators at time $X_t$9, possibly containing both individual-level and cluster-level functions of past history, and choose a reference individual $H_t$0 uniformly at random in $H_t$1 (Shi et al., 2022).

In this setting, the moderated direct log-odds excursion effect for a given $H_t$2 and moderator value $H_t$3 is

$H_t$4

with corresponding odds ratio

$H_t$5

Each probability is taken under a reference distribution for future treatments $H_t$6, such as the original randomization probabilities or a fixed regime $H_t$7 (Shi et al., 2022).

To capture within-cluster interference, the same framework defines a pairwise-indirect log-odds effect by selecting two distinct random indices $H_t$8 and $H_t$9 in a cluster and comparing the log-odds of $A_t$0 when $A_t$1 is assigned treatment versus control, holding $A_t$2’s own treatment at zero:

$A_t$3

with

$A_t$4

This distinction between direct and pairwise-indirect effects is central when the scientific question concerns both individualized treatment effects and spillover within the cluster. It also marks a substantive extension beyond earlier individual-level excursion-effect formulations, because the target estimands explicitly accommodate between-cluster treatment effect heterogeneity and within-cluster interference (Shi et al., 2022).

4. Identification and estimation on the logit scale

Identification follows the same core structure as in the broader causal excursion literature. By analogy to Robins’ G-formula, the required assumptions are consistency, positivity, and sequential ignorability or intervention randomization. In the clustered logit formulation, consistency requires that the observed $A_t$5 equal its potential outcome under the realized cluster treatment history; positivity requires $A_t$6 for all $A_t$7 seen with positive probability; and sequential ignorability requires that, given $A_t$8, $A_t$9 be independent of all future potential outcomes. In MRTs, the mobile-trial randomization is designed to implement exactly such a sequential randomization scheme, possibly depending on $I_t\in\{0,1\}$0 (Shi et al., 2022).

For moderated direct effects, the proposed estimator uses a working logistic model

$I_t\in\{0,1\}$1

where $I_t\in\{0,1\}$2 is a known $I_t\in\{0,1\}$3 function of the moderator and $I_t\in\{0,1\}$4 are “control” covariates introduced to improve precision. The method combines this model with two weights:

$I_t\in\{0,1\}$5

where $I_t\in\{0,1\}$6 is the chosen reference future regime and $I_t\in\{0,1\}$7 is any function of $I_t\in\{0,1\}$8 only, often equal to the marginal $I_t\in\{0,1\}$9 (Shi et al., 2022).

The estimating equation is a weighted, centered estimating-equation system:

$t$0

Once $t$1 is obtained, the moderated log-odds effect at $t$2 is $t$3 and the moderated odds ratio is $t$4 (Shi et al., 2022).

For the pairwise-indirect effect, the analogous working model is

$t$5

with weights $t$6 and a parallel estimating equation. This produces a single inferential architecture for either direct or pairwise-indirect moderated log-odds effects (Shi et al., 2022).

5. Robust estimators in individual-level MRTs

For individual-level MRTs, the recent odds-ratio literature proposes two estimators for the best linear projection parameter $t$7. The first is a doubly robust estimator under Simple Randomization, defined by the condition $t$8, so that $t$9. In this case,

$x$0

where $x$1 if $x$2 or a known weight for $x$3. The estimator uses three nuisance functions:

$x$4

$x$5

$x$6

and solves a projection-adjusted logistic partially linear score equation $x$7 after first fitting $x$8 (Yu et al., 24 Sep 2025).

The central robustness statement is that, under standard regularity, if either $x$9 or $$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$0 is consistently estimated for each $$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$1, then $$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$2. Moreover, if

$$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$3

then

$$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$4

with a sandwich-form variance estimated by plug-in sample analogues. The estimator is described as doubly robust and locally efficient among that class (Yu et al., 24 Sep 2025).

The second estimator addresses general randomization, when $$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$5 depends on $$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$6 beyond $$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$7 and the Simple Randomization condition fails. It introduces an auxiliary association model

$$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$8

fit by solving

$$\mathrm{OR}_t(x)= \frac{P(Y_{t+1}=1\mid A_t=1,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=1,X_t=x,I_t=1)} {P(Y_{t+1}=1\mid A_t=0,X_t=x,I_t=1)/P(Y_{t+1}=0\mid A_t=0,X_t=x,I_t=1)}.$$9

Given $A_t\in\{0,1\}$00, one solves

$A_t\in\{0,1\}$01

for $A_t\in\{0,1\}$02, where the projected estimating function also uses $A_t\in\{0,1\}$03. If the association model $A_t\in\{0,1\}$04 is correctly specified, then $A_t\in\{0,1\}$05 and

$A_t\in\{0,1\}$06

Moreover, under the null $A_t\in\{0,1\}$07, a misspecified association model does not bias the test of no excursion odds ratio, so type I error is controlled. The same construction can be extended to any monotone link functions, including probit or complementary-log-log, by replacing the logit link in both identification and the association model (Yu et al., 24 Sep 2025).

6. Large-sample theory, empirical performance, and applications

Across the logit-scale literature, the estimators are formulated as M-estimators or Z-estimators with sandwich-type asymptotic variance. In the clustered weighted-centered approach, under regularity the pair $A_t\in\{0,1\}$08 is asymptotically normal:

$A_t\in\{0,1\}$09

where $A_t\in\{0,1\}$10 and $A_t\in\{0,1\}$11. In practice, the variance is computed as the empirical “bread $A_t\in\{0,1\}$12 meat $A_t\in\{0,1\}$13 bread” from the sample of clusters, and when $A_t\in\{0,1\}$14 a Mancl–DeRouen small-sample adjustment to the meat is recommended. Under the identifiability assumptions and mild regularity, including a correctly specified linear logit contrast in $A_t\in\{0,1\}$15, $A_t\in\{0,1\}$16 is consistent for the true moderated log-odds parameter $A_t\in\{0,1\}$17 and jointly asymptotically normal at rate $A_t\in\{0,1\}$18. Wald confidence intervals on the log-odds scale are then exponentiated to obtain valid confidence intervals for the odds ratio (Shi et al., 2022).

The earlier relative-risk formulation yielded analogous asymptotics. Under bounded $A_t\in\{0,1\}$19 and $A_t\in\{0,1\}$20, compact parameter space, and uniqueness of the population estimating-equation solution, one obtains

$A_t\in\{0,1\}$21

with a sandwich form for $A_t\in\{0,1\}$22, and a Mancl–DeRouen leverage adjustment with $A_t\in\{0,1\}$23-quantiles was reported to improve coverage in $A_t\in\{0,1\}$24–$A_t\in\{0,1\}$25 settings (Qian et al., 2019). This suggests a continuity of inferential logic across effect scales even though the target estimands differ.

Simulation evidence in the odds-ratio literature was organized around two sets of simulations with $A_t\in\{0,1\}$26 replicates and $A_t\in\{0,1\}$27 decision points per subject. Under Simple Randomization, with $A_t\in\{0,1\}$28, specified models for $A_t\in\{0,1\}$29 and $A_t\in\{0,1\}$30, and implementations varying whether nuisance models were correctly specified or misspecified, $A_t\in\{0,1\}$31 was unbiased and nominal-CI-covering if at least one of $A_t\in\{0,1\}$32 or $A_t\in\{0,1\}$33 was correct, whereas it failed if both were wrong. $A_t\in\{0,1\}$34 was unbiased when $A_t\in\{0,1\}$35 was correct but could fail otherwise, and logistic GEE/GAM procedures were biased whenever their mean model was misspecified. Under general randomization, only $A_t\in\{0,1\}$36 remained consistent; SR, GEE, and GAM were biased (Yu et al., 24 Sep 2025).

The relative-risk precursor reported simulations varying presence or absence of an important moderator $A_t\in\{0,1\}$37, correct or incorrect nuisance specification, and sample sizes $A_t\in\{0,1\}$38 with $A_t\in\{0,1\}$39. The main findings were that the EMEE estimator was consistent for the marginal excursion effect even with misspecified $A_t\in\{0,1\}$40, that naive use of the full-history SNMM-based ECE estimator without including $A_t\in\{0,1\}$41 in $A_t\in\{0,1\}$42 was inconsistent for the marginal effect, and that the small-sample correction restored coverage to approximately $A_t\in\{0,1\}$43–$A_t\in\{0,1\}$44 (Qian et al., 2019). In the clustered logit-scale setting, extensive simulation studies were reported to confirm the theory empirically and to show that the proposed procedure provides consistent point estimator and interval estimates with valid coverage (Shi et al., 2022).

Empirical illustrations anchor the methodology in mobile health. In BariFit, a 45-participant post-bariatric surgery MRT with $A_t\in\{0,1\}$45 daily decision times, the treatment was a daily food-tracking SMS with $A_t\in\{0,1\}$46, and the proximal binary outcome was whether the participant completed a food log that day. The primary marginal analysis with $A_t\in\{0,1\}$47 estimated $A_t\in\{0,1\}$48 on the log-relative-risk scale, with $A_t\in\{0,1\}$49 CI $A_t\in\{0,1\}$50 and $A_t\in\{0,1\}$51, corresponding to $A_t\in\{0,1\}$52 and no statistically detectable proximal effect; moderation by day-in-study, gender, and previous-day logging was also not significant (Qian et al., 2019).

In Drink Less, a 30-day MRT with 349 heavy-drinking adults randomized at 20:00 each day, $A_t\in\{0,1\}$53 indicated that a notification was sent with probability $A_t\in\{0,1\}$54, $A_t\in\{0,1\}$55 included decision-point index, whether the app was opened before 20:00, and whether a notification was sent yesterday, and $A_t\in\{0,1\}$56 indicated that the app was opened in the following hour. Using both SR and GR estimators with $A_t\in\{0,1\}$57 and spline terms where appropriate, the estimated marginal log-OR was $A_t\in\{0,1\}$58 with $A_t\in\{0,1\}$59 CI $A_t\in\{0,1\}$60, corresponding to $A_t\in\{0,1\}$61 with interval $A_t\in\{0,1\}$62. No moderator slope was statistically significant in any of the three moderation analyses, and SR and GR gave virtually identical results because randomization was constant (Yu et al., 24 Sep 2025).

The current methodological picture therefore separates three issues that are sometimes conflated. First, the causal excursion effect is a design-based estimand for sequentially randomized interventions, not merely a regression coefficient. Second, the odds-ratio version is not interchangeable with the relative-risk version, although the two are structurally parallel. Third, clustered MRTs with interference require direct and indirect excursion effects that are not reducible to the standard individual-level estimand. Future work identified in the odds-ratio literature includes nonparametric models for $A_t\in\{0,1\}$63, sample-size formulas for odds-ratio targets, and efficiency-gain strategies via auxiliary covariates (Yu et al., 24 Sep 2025).