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A Cautionary Tale on Integrating Studies with Disparate Outcome Measures for Causal Inference
Robust decision-making increasingly depends on integrating information from diverse sources -- a practice commonly referred to as data integration. By harnessing complementary datasets, researchers can improve the accuracy, generalizability, and efficiency of statistical inference bareinboim2016causal. In the realm of causal inference, data integration has emerged as a central focus, recently cited among the top ten priorities for advancing the field mitra2022future. This surge of interest reflects its wide-ranging utility: from generalizing or transporting evidence degtiar2023review, parikh2024we, huang2024towards, to heterogeneous causal effect estimation brantner2023methods, boosting statistical efficiency rosenman2023combining, and mitigating bias Kallus2018.
However, in many real-world scenarios, various data sources may capture outcomes that, while related, are not identical to those measured in the trial. For example, in studies on medications for opioid use disorder (MOUD), the intensity of withdrawal symptoms can be measured using two different scales: the Clinical Opiate Withdrawal Scale (COWS) and the Subjective Opiate Withdrawal Scale (SOWS) wesson2003clinical, handelsman1987two. In the XBOT trial that compared the effectiveness of injection naltrexone to sublingual buprenorphine in terms of reducing risks of returning to regular opioid use, withdrawal symptoms were measured using SOWS lee2018comparative. However, the POATS study, which compared the effectiveness of adding counseling to sublingual buprenorphine treatment, used COWS to measure the strength of withdrawal symptoms weiss2010multi. Despite the differences in outcome measures, researchers might wish to leverage the POATS study to improve the precision of treatment effect estimates in the XBOT trial (or vice versa). This raises an important question: when can integration of primary study and auxiliary data with disparate outcome measures yield efficiency gains for causal effect estimates if neither study has observation of both outcome measures on the same group of individuals?
\paragraph{Contributions.} Our paper addresses this question by examining scenarios in which neither the trial nor the auxiliary data records both outcome measures on the same set of individuals.
The paper is organized as follows. Section (ref) introduces the notation, setup, and standard assumptions. Section (ref) presents the key structural assumption linking primary and auxiliary outcomes, along with three scenarios that reflect varying degrees of prior knowledge about this relationship. Sections (ref)--(ref) contains our main theoretical contributions: semiparametric efficiency bounds under each scenario, as well as worst-case bounds on finite-sample estimation errors. In Section (ref), we apply these methods to estimate the causal effect of medications for opioid use disorder (MOUD) on withdrawal severity, using SOWS (from the XBOT trial) and COWS (from the POAT study). Section (ref) concludes with a summary of key findings, limitations, and directions for future research. Appendix (ref) presents simulation results evaluating estimator performance across varying sample sizes and dimensions. Additional theoretical discussion and proofs are provided in Appendices (ref), (ref), and (ref).
We briefly review four bodies of literature related to our work: (i) data integration in causal inference, (ii) meta-analysis, (iii) data harmonization, and (iv) surrogate outcomes.
\paragraph{Data Integration for Causal Inference.} Data integration has emerged as a central focus in causal inference, recently cited among the top priorities for advancing the field mitra2022future. It supports a wide range of goals, including generalizing evidence across populations degtiar2023review, pearl2015generalizing, parikh2024we, huang2024towards, estimating heterogeneous effects brantner2023methods, boosting efficiency li2023efficient, and mitigating bias Kallus2018, morucci2023double. Recent methods improve efficiency by combining auxiliary datasets while controlling bias, such as James-Stein shrinkage rosenman2023combining, semiparametric estimators yang2020improved, bias correction Kallus2018,YangDing2020, and Bayesian borrowing lin2023many. However, these approaches typically assume consistent measures -- including outcomes -- across datasets.
\paragraph{Meta-Analysis and Evidence Synthesis.} When outcomes differ, naïve pooling can induce substantial bias van2011combining. Early evidence synthesis methods, such as standardizing outcomes murad2019continuous, deeks2019analysing, rely on strong equivalence assumptions. Traditional meta-analyses use heuristics like dichotomization or normalization deeks2019analysing, assuming commensurability across studies murad2019continuous. More sophisticated approaches jointly model multiple outcomes, using multivariate Bayesian methods bujkiewicz2016bayesian or multi-task learning analogs zhang2018overview. These frameworks exploit known outcome dependencies or co-measurement of outcomes to synthesize information while allowing outcome-specific variation.
\paragraph{Data Harmonization.} Data harmonization methods are a set of tools that aim to equate measures across data sources to facilitate data integration. These methods typically align heterogeneous outcomes through co-calibration nance2017co or latent constructs snavely2014latent. Bridge studies, where multiple outcomes are measured on the same set of individuals, can estimate mappings between outcome measures, while latent variable models treat observed outcomes as noisy indicators of a shared construct. These approaches typically require both outcome measurements for the same individual and introduce additional modeling assumptions.
\paragraph{Leveraging Surrogate Outcomes.} Another relevant literature is on data integration methods leveraging studies with surrogate outcomes. For instance, athey2019surrogate and ghassami2022combining combine experimental data with short-term outcome measures with an observational study where long-term outcome is measured to yield a consistent estimate of the long-term treatment effect. Surrogate indices that aggregate multiple proxies can substantially improve efficiency ghassami2022combining, but rely on strong structural assumptions about the proxy–outcome relationship.
Existing methods generally require at least one dataset with measurement of primary and surrogate outcomes on the same set of individuals -- an assumption often violated in practice and one that motivates our work.
\paragraph{Setup and Notations.} We consider two studies: a primary study (\(S = 0\)) and an auxiliary study (\(S = 1\)). The primary study observes the outcome of interest \(Y\), while the auxiliary study observes a related but distinct outcome \(W\). Crucially, \(Y\) and \(W\) are never observed for the same individual. In both studies, we observe treatment \(T \in \{0,1\}\) and covariates \(X\). Let \(Y(t)\) and \(W(t)\) denote the potential outcomes under treatment \(T = t\). To unify notation, define the observed outcome as \( V := (1 - S)Y + S W, \) and the observed data as \( O := (X, S, T, V). \) We let \(\mathcal{S}_n = \{O_1, \ldots, O_n\}\) denote a sample of \(n\) units, with \(n_0\) and \(n_1\) representing the number of units in the primary and auxiliary studies, respectively.
For any (random) function \(f\), let \(\mathbb{E}[f(A)]\) denote the expectation, \(\mathcal{P}_n(f(A)) = \frac{1}{n} \sum_{i=1}^n f(A_i)\) the empirical average, and \(\mathcal{P}(f(A)) = \int f(a)\, dP(a)\) the population average treating \(f\) as fixed. Note that \(\mathbb{E}[f(A)]\) integrates over randomness in both \(A\) and \(f\), while \(\mathcal{P}(f(A))\) treats \(f\) as fixed. We also define the \(L^q(P)\) norm as \(\|f\|_q = \left( \int |f(o)|^q\, dP(o) \right)^{1/q}\). Futher, for compactness, we write \(\mu_A(B = b) := \mathbb{E}[A \mid B = b]\) to denote the conditional expectation of \(A\) given \(B = b\), and \(\nu^t_A(B = b) := \mathbb{E}[A(t) \mid B = b]\) for the conditional mean of the potential outcome \(A(t)\).
Our goal is to estimate the conditional average treatment effect (CATE): \(\tau_0(x) := \nu^1_Y(X = x) - \nu^0_Y(X = x)\), and the average treatment effect (ATE): \(\tau_0 := \nu^1_Y - \nu^0_Y\), both defined with respect to the primary outcome \(Y\).
\paragraph{Assumptions & Identification} We make the following assumptions:
We assume the following structural models for the potential outcomes:
where $\theta(X)$ and $\phi(X)$ are the treatment effect functions for $Y$ and $W$, respectively. These formulation is commonly used in the causal inference literature robinson1988root, chernozhukov2018double, hahn2020bayesian, rudolph2023improving. From Equation (ref), it follows that the CATE in the primary population is: \( \tau_0(x) = \mathbb{E}[Y(1) - Y(0) \mid X = x] = \theta(x). \label{eq: theta_est} \) By Assumptions (ref) and (ref), the potential outcome means $\nu^t_Y(X = x)$ are identified by observed data as: \( \nu^t_Y(X = x) = \mu_Y(X = x, T = t). \) Hence, the CATE is identified as: \( \tau_0(x) = \mu_Y(X = x, T = 1) - \mu_Y(X = x, T = 0). \)
\paragraph{Influence Function.} Let $\eta$ denote the collection of nuisance parameters, specifically $\eta = \{\mu_V(X,S), \mu_T(X,S), \mu_S(X)\}$. Let $\theta_0$ be the true parameter of interest, and $\eta_0$ the true nuisance parameters governing the data-generating process. For any regular, consistent, and asymptotically linear estimator $\hat{\theta}$ of $\theta_0$, there exists a function $\psi$ -- called the influence function -- such that we decompose the estimation error $\hat{\theta} - \theta_0$ using the von Mises expansion as: \[ \hat{\theta} - \theta_0 = \underbrace{(\mathcal{P}_n - \mathcal{P})\psi(O; \theta_0, \eta_0)}_{M_1} - \underbrace{\mathcal{P}\left[\psi(O; \theta_0, \hat{\eta}) - \psi(O; \theta_0, \eta_0)\right]}_{M_2(\hat{\eta})} + M_3(\hat{\eta}) \] where $\mathcal{P}[\psi(O; \theta_0, \eta_0)] = 0$ tsiatis2006semiparametric, kennedy2016semiparametric. Here, (i) the first term, $M_1$, represents sampling variability resulting in asymptotic variance -- capturing the first-order behavior of $\hat{\theta}$ and reflects its asymptotic linearity ichimura2022influence, kennedy2016semiparametric; (ii) the second term, \(M_2(\hat{\eta})\), captures bias due to finite sample estimation of nuisance functions; (iii) the third term, $M_3(\hat{\eta})$, accounts for remaining higher-order approximation error that converges to 0 in probability at rate faster than $\sqrt{n}$.
By the Central Limit Theorem and the Slutsky theorem, the estimator is asymptotically normal (provided Donsker condition holds or sample splitting is used): \[ \sqrt{n}(\hat{\theta} - \theta_0) \rightsquigarrow \mathcal{N}\left(0, \mathbb{E}[\psi(O; \theta_0, \eta_0)\psi(O; \theta_0, \eta_0)^T]\right). \] The asymptotic variance of $\hat{\theta}$ is thus determined by the variance of the influence function and can be consistently estimated via the empirical variance of the estimated influence function, $\hat{\psi}$.
\paragraph{Efficient Influence Function.} The influence function depends on the values of $\theta$ and $\eta$, although we suppress this dependency for notational convenience. Among all influence functions corresponding to regular, asymptotically linear estimators of $\theta_0$, the efficient influence function (EIF), denoted $\psi^*$, achieves the smallest possible asymptotic variance. This minimal variance -- known as the semiparametric efficiency bound -- is given by: \( \mathbb{E}[\psi^*(O; \theta_0, \eta_0)\psi^*(O; \theta_0, \eta_0)^T], \) and represents the best achievable precision for unbiased estimation tsiatis2006semiparametric, newey1990semiparametric.
\paragraph{Procedure to Derive EIF.} Consider the log-likelihood $\mathcal{L}(O; \theta, \eta)$ of observed data with parameter of interest $\theta$ and nuisance parameters $\eta$, maximized at $(\theta_0, \eta_0)$. We define the score functions with respect to $\theta$ and $\eta$ as $R_\theta(O; \theta_0, \eta_0) = \left. \frac{\partial \mathcal{L}}{\partial \theta} \right|{\theta_0, \eta_0}$ and $R_\eta(O; \theta_0, \eta_0) = \left. \frac{\partial \mathcal{L}}{\partial \eta} \right|{\theta_0, \eta_0}$, respectively. $R_\theta$ reflects the sensitivity of the likelihood to $\theta$. However, it may also be sensitive to $\eta$. Projecting it orthogonally to the space spanned by $R_\eta$ isolates the component of information unique to $\theta$. The efficient score function is the residual of $R_\theta$ after projecting out components in the linear span of $R_\eta$: \[ R^*(O; \theta_0, \eta_0) = R_\theta - \Pi\big(R_\theta ,|, \Lambda_{\eta}\big), \] where $\Pi\big(R_\theta ,|, \Lambda_{\eta}\big) = \mathbb{E}\left[R_\theta R_\eta^T\right] \left\{\mathbb{E}\left[R_\eta R_\eta^T\right]\right\}^{-1} R_\eta$ and arguments $(O; \theta_0, \eta_0)$ are suppressed for brevity. The efficient influence function is given by \( \psi^* = \left\{ \mathbb{E}\left[R^* (R^*)^{T} \right] \right\}^{-1} R^*. \) This influence function achieves the semiparametric efficiency bound and serves as the optimal estimating function for $\theta$ under the given model. For further discussion and derivation, we refer readers to tsiatis2006semiparametric.
To leverage auxiliary data for estimating treatment effects on the primary outcome $Y$, we must establish a relationship between $Y$ and the auxiliary outcome $W$. We posit the following structural assumption that provides a foundation -- or “license” -- for incorporating $W$ into the analysis:
This assumption is plausible in settings where $W$ serves as a meaningful proxy for $Y$. For instance, in biomedical studies, $W$ might represent a surrogate endpoint (e.g., a biomarker) that reflects the underlying disease progression captured by $Y$ weir2006statistical. In such cases, prior studies or mechanistic understanding can inform how changes in $W$ relate to changes in $Y$.
To explore the range of identifiability and efficiency in leveraging auxiliary data, we consider three increasingly weaker assumptions about prior knowledge about $\alpha(X)$ and $\beta(X)$:
Assumption (ref) represents the strongest assumption, and is most tenable in domains with well-characterized mechanistic knowledge -- such as certain areas of biology, pharmacology, or engineering -- where $\alpha(X)$ and $\beta(X)$ are grounded in empirical studies or physical theory puniya2018mechanistic, parikh2023effects. In such contexts, auxiliary outcomes can be confidently incorporated using known mappings to the primary outcome. Assumption (ref) relaxes this requirement by assuming only the baseline shift $\beta(X)$ is known. This is common in applications where historical data or expert knowledge informs baseline trends, but the strength of association (i.e., scaling) between $Y$ and $W$ varies across populations or settings. Such partial knowledge arises frequently in social sciences or public health handelsman1987two. Assumption (ref)) is the most general and aligns with many real-world scenarios where no prior information is available about the relationship between $Y$ and $W$. This assumption allows maximum flexibility, but also introduces the greatest challenge in using an auxiliary study.
These assumptions represent a spectrum of tradeoffs between realism and statistical precision. Stronger assumptions enable tighter and more efficient estimation but rely more heavily on prior knowledge. We make this tradeoff explicit in Sections (ref), (ref) and (ref). Ultimately, the appropriate assumption set depends on the context and credibility of available domain knowledge.
\paragraph{Function Class Complexity Assumption.} We make additional assumptions about the complexity of functions in (ref):
This assumption imposes regularity conditions on the function classes involved in the decomposition of \(\mu_Y(X)\) into \(\alpha(X)\) and \(\beta(X)\). Specifically, it ensures that the combined complexity of \(\alpha\) and \(\beta\), measured via covering number bounds, does not exceed that of \(\mu_Y\). This is a mild and natural requirement: if the auxiliary outcome \(W\) is informative about \(Y\), then the residual mapping captured by \(\alpha(X)\) is expected to be simpler than modeling \(\mu_Y(X)\) directly. In this sense, Assumption (ref) reflects a form of functional regularization, where using a predictive surrogate reduces the effective complexity of the learning task.
Now, we derive the efficient bounds under each of the following three assumptions (ref), (ref), and (ref), and investigate if and when data integration yields semiparametric efficiency gains. Throughout this section, we assume that assumptions (ref) to (ref) hold.
Recall $\psi^*(O; \theta_0, \eta_0) = \left\{\mathbb{E}[R^*(O; \theta_0, \eta_0) R^*(O; \theta_0, \eta_0)^T]\right\}^{-1} R^*(O; \theta_0, \eta_0),$ and the semiparametrically efficient asymptotic variance (i.e., efficiency bound) is equal to $\left\{\mathbb{E}[R^*(O; \theta_0, \eta_0) R^*(O; \theta_0, \eta_0)^T]\right\}^{-1}$, we only present the efficient score function $R^*$ instead of the EIF $\psi^*$. However, note that deriving the EIF from $R^*$ is straightforward in our context. In our case, $P(O=o ; \theta, \eta) = P(X=x)P(S=s \mid X=x)P(T=t \mid X=x, S=s)P(V=v \mid T=t, X=x, S=s)$ and $\mathcal{L}(O; \theta, \eta) = \log P(O=o ; \theta, \eta)$.
First, we derive the efficiency bound for the semiparametrically efficient estimator that only uses the primary study. We use this result as a base case to compare the efficiency bounds for the data integration-based estimators. This efficiency bound is akin to the one derived in robinson1988root.
Now, we derive the efficiency bound that leverages auxiliary data under (ref).
Next, we derive the efficiency score and the efficiency bound for a case when (ref) holds.
The proofs and results in Theorems (ref) to (ref) are a direct consequence of following the procedure to derive EIF described in Section (ref) and are provided in Appendix (ref).
Now, we present the ATE estimation under assumption (ref). We use the efficient score $R^*_a$ to guide the estimation of $\theta_0$ using the property that $\mathbb{E}[R^*_a(O;\theta_0,\eta_0)] = 0$. Recall, that $R^*_a(O;\theta_0,\eta_0) = S \Delta_1 + (1-S) \Delta_0$. Assuming an unbiased and consistent estimate of the nuisance parameter $\hat{\eta}$, a solution to $\frac{1}{n_0} \sum_i R^*_a(O_i;\theta,\hat{\eta})$ -- denoted by $\hat{\theta}_a$ -- is an unbiased and consistent estimate of $\theta_0$. Let $r_A(B) := A - \mu_A(B)$ denote the residual of random variable $A$ after regressing $A$ on $B$. Then, the estimator $\hat{\theta}_a$ is given by: \[ \hat{\theta}_a = \frac{\sum_i \left( (1-S_i)\frac{\hat{r}_Y(X_i,0)\hat{r}_T(X_i,0)}{\hat{\sigma}^2_Y} + S_i\frac{\hat{r}_W(X_i,1)\hat{r}_T(X_i,1)}{\alpha(X_i)\hat{\sigma}^2_W} \right)}{\sum_i\left((1-S_i)\frac{\hat{r}_T^2(X_i,0)}{\hat{\sigma}^2_Y} + S_i\frac{\hat{r}_T^2(X_i,1)}{\alpha^2(X_i)\hat{\sigma}^2_W} \right)} \]
\paragraph{Misspecification Bias under (ref):} We showed the efficiency bound under three varied assumptions and our results highlighted that efficiency gain is only feasible under the strongest assumption. Now, we investigate the cost of making the wrong assumption i.e. what happens if we assume (ref) but $\alpha$ is misspecified. Let $\alpha^\star$ denote the true $\alpha$ and $\alpha_{mis}$ denote a misspecified $\alpha$.
In cases when $\alpha$ is unknown (i.e. (ref) and (ref)), it is not feasible to yield efficiency gains by leveraging auxiliary data. However, consider the estimator for ATE only using primary data \[ \hat{\theta}_{0} = \frac{\sum_i (1-S_i)[ (\hat{r}_Y(X_i,0))(\hat{r}_T(X_i,0)) ]}{\sum_i (1-S_i)[ (\hat{r}_T(X_i,0))^2 ]}. \]
This estimator can be modified, under (ref), to use the auxiliary data to potentially have finite sample benefits. One natural approach to leveraging auxiliary data is the following two-stage estimator: in the first stage, we estimate the auxiliary regression \(\mu_W(X,1) = \mathbb{E}[W | X, S=1]\) using the auxiliary data and we then use this estimated function to predict \(\hat{\mu}_W(X,0)\) for units in the primary data. In the second stage, we estimate $\mu_Y$ as: \( \hat{\mu}_{Y,b}(X,0) = \hat{\alpha}(X)\hat{\mu}_W(X,0) + \hat{\beta}(X) \) where \( \hat{\alpha}, \hat{\beta} \in \left[ \arg \min_{ \alpha, \beta \in \mathcal{A}, \mathcal{B}} \frac{1}{n_0} \sum_i (1-S_i)\left( Y_i - \alpha(X_i) \hat{\mu}_W(X_i,0) - \beta(X_i) \right)^2 \right]. \) The resulting fitted function \(\hat{\mu}_{Y,b}(X,0)\) combines both sources of information and provides a data-adaptive estimator of the conditional mean outcome in the primary population. This approach is akin to adjusting for the prognostic or benefit score along with the vector of covariates liao2025prognostic. Thus, the resulting estimator leveraging the auxiliary data is given as: \[ \hat{\theta}_b = \frac{\sum_i (1-S_i)[(\hat{r}_{Y,b}(X_i,0))(\hat{r}_T(X_i,0)) ]}{\sum_i (1-S_i)[ (\hat{r}_T(X_i,0))^2 ]} . \]
\paragraph{Quantifying Finite-Sample Risk.} Asymptotically, if nuisance estimators $\hat{\eta}$ belong to a Donsker class or are fit using sample splitting, the $M_2$ and $M_3$ vanishes asymptotically at a rate faster than \(\sqrt{n}\). However, in finite samples, they contribute non-negligibly to estimation error. We focus on $M_2(\hat{\eta}) = \mathcal{P}(\psi(O; \theta_0, \hat{\eta}) - \psi(O; \theta_0, \eta_0)),$ which depends on the accuracy of nuisance function estimates. For cross-fitted estimators, we have: \( |M_2(\hat{\eta})| = o_p\left(n^{-1/2} \left( \|\mu_Y - \hat{\mu}_Y\| \cdot \|\mu_T - \hat{\mu}_T\| + \theta_0\, \|\mu_T - \hat{\mu}_T\|^2 \right) \right). \) Since the only difference between \(\hat{\theta}_0\) and \(\hat{\theta}_b\) lies in the choice of outcome regression, smaller \(\|\mu_Y - \hat{\mu}_Y\|\) directly translates into precise estimates.
\paragraph{Characterizing Finite Sample Gains.} Theorem (ref) demonstrates that one may not even achieve finite sample gains when leveraging auxiliary data. The two-stage estimator \(\hat{\mu}_{Y,b}\) can outperform the direct regression estimator \(\hat{\mu}_{Y,0}\) only when certain structural and sample size conditions are met. Qualitatively, gains arise when leveraging the auxiliary data allow for decomposition of \(\mu_Y(X)\) into less complex functions. Additionally, leveraging auxiliary data helps only if \(\mu_W(X)\) can be estimated accurately -- that is, when the auxiliary sample size \(n_1\) is sufficiently large relative to the primary sample size \(n_0\). Importantly, when the auxiliary outcome \(W\) is highly predictive of the primary outcome \(Y\) -- that is, when \(\text{Cov}(Y, W \mid X)\) is large -- the function \(\mu_Y(X)\) can be well-approximated by \(\mu_W(X)\) then the function \(\alpha(X)\) captures only residual structure and tends to be significantly simpler than \(\mu_Y(X)\) itself, implying that the entropy exponent \(\omega_\alpha\) is relatively much smaller than \(\omega\). Quantitatively, finite-sample improvement occurs when \( n_0^{\frac{1}{2 + \omega} - \frac{1}{2 + \omega_\alpha}} + \left(n_0/n_1\right)^{\frac{1}{2 + \omega}} < 1. \) The first term captures the gain from replacing the full function class \(\mathcal{M}_Y\) with a lower-complexity class \(\mathcal{A}\), and the second term reflects the accuracy of estimating \(\mu_W(X)\) from the auxiliary data. Gains are most pronounced when \(\omega_\alpha \ll \omega\) (i.e., \(\alpha(X)\) is much simpler than \(\mu_Y(X)\)) and when \(n_1 \gg n_0\) (i.e., we have ample auxiliary data). Our characterization formally supports the intuition that structural assumptions and additional data may result in finite sample gains.
We apply our framework to compare the effectiveness of extended-release naltrexone (XR-NTX) and buprenorphine-naloxone (BUP-NX) in reducing opioid withdrawal symptoms between 10 and 12 weeks after treatment initiation. We begin by describing the primary and auxiliary datasets and the causal quantity of interest, followed by estimates obtained under three approaches: (i) using only primary data, (ii) incorporating auxiliary data with known outcome linkage (Assumption (ref)), and (iii) incorporating auxiliary data under partial knowledge of the link (Assumption (ref)).
We evaluate the comparative effectiveness of XR-NTX (\(T = 1\)) versus BUP-NX (\(T = 0\)) in reducing withdrawal symptom severity, as measured by the worst SOWS score in the fourth week (\(Y\)), among participants in the XBOT trial. In the auxiliary POAT study, withdrawal severity is measured on the COWS scale during the same period (\(W\)). We use a common set of covariates assessed in both trials (\(X\)). Further, we only considered patients for whom we observed the outcomes -- our study excluded individuals for whom treatment was not initiated or who dropped out before our outcome window.
To harmonize the two scales, we derive the transformation coefficient \(\alpha\) from published clinical thresholds. According to wesson2003clinical, COWS ranges of 5–12, 13–24, 25–36, and >36 correspond to mild, moderate, moderately severe, and severe withdrawal, respectively. Similarly, handelsman1987two defines SOWS ranges of 1–10, 11–15, 16–20, and 21–30 for the same categories. Assuming both scales share a zero point (no withdrawal), we align category midpoints and estimate a linear mapping \(Y = \alpha W + \varepsilon\), yielding \(\alpha = 0.61\) and intercept \(\beta = 0\). Figure (ref)(a) visualizes this relationship. We assume \(\alpha\) is constant across covariate values \(X\), and interpret lower values of both \(Y\) and \(W\) as indicating better outcomes. We then apply the three estimators introduced in Section (ref) and (ref): \(\hat{\theta}_0\) (primary data only), \(\hat{\theta}_b\) (auxiliary data, unknown \(\alpha\)), and \(\hat{\theta}_a\) (auxiliary data, known \(\alpha\)). As shown in Figure (ref)(b), $\hat{\theta}_0$ and $\hat{\theta}_b$ suggest that XR-NTX and BUP-NX are almost equally effective. However, $\hat{\theta}_a$ suggests BUP-NX is marginally more effective in lowering withdrawal symptoms compared to XR-NTX. Specifically: (i) \(\hat{\theta}_0 = -0.18\) (95% CI width: 1.52), (ii) \(\hat{\theta}_b = 0.42\) (95% CI width: 0.65), and (iii) \(\hat{\theta}_a = -1.08\) (95% CI width: 1.41). While \(\hat{\theta}_a\) achieves a statistically significant result, it relies on the correctness of the assumed \(\alpha\). To assess robustness, we conduct a sensitivity analysis by varying \(\alpha\) within \(\pm50\%\) of the estimated value, i.e., \(\alpha \in [0.31, 0.92]\), assuming the linear form remains valid. Figure (ref)(c) displays the resulting \(\hat{\theta}_a\) estimates across this range. Although the point estimates vary -- from \(0.50\) to \(0.33\) -- they consistently favor BUP-NX over XR-NTX. However, for $\alpha > 0.75$, the 95% confidence intervals include zero.
Summary. This paper presents a principled framework for integrating primary and auxiliary datasets with non-overlapping, disparate outcomes to improve efficiency in causal effect estimation. We focus on settings where the primary outcome is never jointly observed with the auxiliary outcome, and we introduce a structural assumption that links the two. Building on this, we define three scenarios reflecting varying levels of prior knowledge about outcome relationship and derive semiparametric efficiency bounds under each. Our findings show that efficiency gains are guaranteed only under the strongest assumptions, when the linking equation is fully known. In contrast, under weaker assumptions, asymptotic efficiency is not ensured. However, finite-sample improvements are still possible, particularly when the auxiliary outcome is highly predictive of the primary outcome. These benefits taper off as the primary sample size increases, highlighting the limitations of auxiliary data in isolation. We support our theoretical results with both simulations and a case study estimating the effect of medications for opioid use disorder (MOUD) on withdrawal severity. Here, we combine data from the XBOT trial (SOWS scale) and the POAT study (COWS scale), demonstrating the framework’s practical utility.\\ Limitations & Future works. Our analysis and results in this paper depend on the structural assumption between $Y$ and $W$. Moving ahead, we will focus on making our results more general by relaxing this assumption. Further, we will focus on incorporating a third “bridge” dataset, where both outcomes are observed, which could help relax strong assumptions and expand the conditions under which efficiency gains are possible. We will also explore relaxing the assumption of conditional study exchangeability, extending the framework to accommodate discordance in treatments and covariates across studies, and generalizing the linking structure beyond linear relationships to better capture complex dependencies.