SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $\eta_{k,L_1}$  denotes the data-adaptive weights;  $\widehat{\xi}_{T,i}(a)$  and  $\widehat{\xi}_{k,i}(a)$  are the estimated influence functions for the target and source site estimators (see Appendix E.3 for the exact form of the influence functions). The estimated difference  $\widehat{\delta}_k = \widehat{\mu}_{a,k} - \widehat{\mu}_{a,T}$  quantifies the bias between the estimate from source  $k\in S$  and the estimate from the target  $T$ . The tuning parameter  $\lambda$  determines the penalty imposed on source site estimates and in practice, is chosen via cross-validation. Specifically, we create a grid of values of  $\lambda$  and iteratively train and evaluate the model using different  $\lambda$  values, selecting the one with the lowest average validation error after multiple sample splits.
We estimate the variance of  $\widehat{\mu}_{a,G}$  using the estimated influence functions for  $\widehat{\mu}_{a,T}$  and  $\widehat{\mu}_{a,k}$ . By the central limit theorem,  $\sqrt{N} (\widehat{\mu}_{a,G} - \bar{\mu}_{a,G})\overset {d}{\to}\mathcal{N}(0,\Sigma)$ , where  $\Sigma = E\{\sum_{k\in \mathcal{K}}\bar{\eta}_k\xi_{k,i}(a)\}^2$  and  $\bar{\eta}_k$  denote the limiting values of  $\widehat{\mu}_{a,G}$  and  $\hat{\eta}_k$  respectively. The standard error of  $\widehat{\mu}_{a,G}$  is estimated as  $\sqrt{\widehat{\Sigma}/N}$  where  $\widehat{\Sigma} = N^{-1}\sum_{k\in \mathcal{K}}\sum_{i=1}^{n_k}\{\widehat{\eta}_k\widehat{\xi}_{k,i}(a)\}^2$ . A two-sided  $(1-\alpha)\times 100\%$  confidence interval for  $\mu_{a,G}$  is
$$
\widehat {\mathcal {C}} _ {\alpha} = \left[ \widehat {\mu} _ {a, G} - \sqrt {\widehat {\Sigma} / N} \mathcal {Z} _ {\alpha / 2}, \quad \widehat {\mu} _ {a, G} + \sqrt {\widehat {\Sigma} / N} \mathcal {Z} _ {\alpha / 2} \right], \tag {10}
$$
where  $\mathcal{Z}_{\alpha /2}$  is the  $1 - \alpha /2$  quantile for a standard normal distribution.
5 Theoretical Guarantees
In this section, we first establish the theoretical properties of the site-specific estimators constructed with the multiply robust model-mixing approach. Define  $\overline{\pi}_{a,k}^{j}$ ,  $\overline{m}_{a,k}^{l}$ ,  $\overline{\tau}_{a,k}$  and  $\overline{\zeta}_k$  as non-stochastic
functionals that the corresponding estimators  $\widehat{\pi}_{a,k}^{j}$ ,  $\widehat{m}_{a,k}^{l}$ ,  $\widehat{\tau}_{a,k}$  and  $\widehat{\zeta}_k$  converge to for  $k\in \mathcal{K}^1$ . That is,
$$
\| \widehat {\pi} _ {a, k} ^ {j} - \overline {{\pi}} _ {a, k} ^ {j} \| = o _ {p} (1), \quad \| \widehat {m} _ {a, k} ^ {l} - \overline {{m}} _ {a, k} ^ {l} \| = o _ {p} (1), \quad \| \widehat {\tau} _ {a, k} - \overline {{\tau}} _ {a, k} \| = o _ {p} (1), \quad \| \widehat {\zeta} _ {k} - \overline {{\zeta}} _ {k} \| = o _ {p} (1).
$$
As shown in Lemmas E.1 and E.2 in Appendix E.2, the  $L_{2}$  risks of the model-mixing estimators  $\widehat{\pi}_{a,k}$  and  $\widehat{m}_{a,k}$  are bounded by the smallest risks of all candidate models plus a remainder term that vanishes at a faster rate than the risks themselves. Leveraging Lemmas E.1 and E.2, we show the consistency and asymptotic normality of the site-specific estimators in Theorem 2. The proofs are given in the Appendix E.4.
Theorem 2. Suppose that the conditions in Lemmas E.1 and E.2 hold, and for  $k \in \mathcal{K}$  that  $\widehat{\pi}_{a,k}^{j}, \widehat{m}_{a,k}^{l}, \widehat{\zeta}_{k}, \widehat{\tau}_{a,k}, \bar{\pi}_{a,k}^{j}, \bar{m}_{a,k}^{l}, \bar{\zeta}_{k}$  and  $\bar{\tau}_{a,k}$  are uniformly bounded for all treatment models  $j \in \mathcal{J}$  and for all outcome models  $l \in \mathcal{L}$ . Consider the following conditions:
(B1)  $\overline{\pi}_{a,k}^{j} = \pi_{a,k}$  for some  $j\in \mathcal{J}$
(B2)  $\overline{m}_{a,k}^{l} = m_{a,k}$  for some  $l\in \mathcal{L}$
(C1)  $\overline{\zeta}_k = \zeta_k$
(C2)  $\overline{\tau}_{a,k} = \tau_{a,k}$ .
Then, under Assumptions (A1) - (A7), and if one of (B1) or (B2) and one of (C1) or (C2) hold,
$$
\left\| \widehat {\mu} _ {a, k} - \mu_ {a, T} \right\| = O _ {p} \left(n ^ {- 1 / 2} + \| \widehat {\pi} _ {a, k} - \pi_ {a, k} \| \| \widehat {m} _ {a, k} - m _ {a, k} \| + \| \widehat {\zeta} _ {k} - \zeta_ {k} \| \| \widehat {\tau} _ {a, k} - \tau_ {a, k} \|\right). \tag {11}
$$
Further, if the nuisance estimators satisfy the following convergence rate
$$
\left\| \widehat {m} _ {a, k} - m _ {a, k} \right\| \left\| \widehat {\pi} _ {a, k} - \pi_ {a, k} \right\| = o _ {p} (1 / \sqrt {n}), \quad \left\| \widehat {\zeta} _ {k} - \zeta_ {k} \right\| \left\| \widehat {\tau} _ {a, k} - \tau_ {a, k} \right\| = o _ {p} (1 / \sqrt {n}), \tag {12}
$$
then  $\sqrt{n} (\widehat{\mu}_{a,k} - \mu_{a,T})$  asymptotically converges to a normal distribution with mean zero and asymptotic variance equal to the semiparametric efficiency bound.
In Theorem 3, we establish the consistency and asymptotic normality of the federated global estimator. These properties are attained without requiring consistency in the site-specific estimators from the source sites, a consequence of the adaptive weighting method employed. We further delve into conditions in which leveraging information from source sites can enhance the efficiency of the estimator constructed solely with the target data. The proofs are given in the Appendix E.5.
Theorem 3. Under Assumptions (A1) - (A7) and the regularity conditions specified in the Appendix E.5, the federated global estimator of  $\Delta_T$ , given by  $\widehat{\Delta}_G = \widehat{\mu}_{1,G} - \widehat{\mu}_{0,G}$ , is consistent and asymptotically normal,
$$
\sqrt {N / \widehat {\mathcal {V}}} \left(\widehat {\Delta} _ {G} - \Delta_ {T}\right) \xrightarrow {d} \mathcal {N} (0, 1), \tag {13}
$$
with the variance estimated consistently as  $\widehat{\mathcal{V}}$ . The variance of  $\widehat{\Delta}_G$  is no larger than that of the estimator based on target data only,  $\widehat{\Delta}_T = \widehat{\mu}_{1,T} - \widehat{\mu}_{0,T}$ . Further, if there exist some source sites with consistent estimators of  $\Delta_T$  and satisfy conditions specified in the Appendix E.5, the variance of  $\widehat{\Delta}_G$  is strictly smaller than  $\widehat{\Delta}_T$ .
We evaluate the finite sample properties of five different estimators: (i) an augmented inverse probability weighted (AIPW) estimator using data from the target site only (Target), (ii) an AIPW estimator that weights each site proportionally to its sample size (SS), (iii) an AIPW estimator that weights each site inverse-proportionally to its variance (IVW), (iv) an AIPW estimator that weights each site with the  $L_{1}$  weights defined in (9) (AIPW- $L_{1}$ ), and (v) a multiply robust estimator with the
<sup>1</sup>When  $k = T$ , the site-specific estimator takes the form in (4) and we only specify the outcome and treatment models. To establish the consistency and asymptotic normality of the target site-specific estimator, we require one of the Assumptions (B1) and (B2) in Theorem 3 to hold. Then (11) is simplified since we only need to consider the convergence rate of  $\| \widehat{m}_{a,k} - m_{a,k} \| \| \widehat{\pi}_{a,k} - \pi_{a,k} \|$ .
$L_{1}$  weights defined in (9)  $(\mathrm{MR} - L_{1})$ . Across different settings, we examine the performance of each estimator in terms of mean absolute error, root mean square error, and coverage and length of  $95\%$  confidence intervals (CI) across 500 simulations.
We consider a total of five sites and fix the first site as the target site with a relatively small sample size of 300. The source sites have larger sample sizes of  $\{500, 500, 1000, 1000\}$ . We model heterogeneity in the covariate distributions across sites with skewed normal distributions and varying levels of skewness in each site,  $X_{kp} \sim \mathcal{SN}\left(x; \Xi_{kp}, \Omega_{kp}^2, \gamma_{kp}\right)$ , where  $k \in \{1, \dots, 5\}$  indexes each site and  $p \in \{1, \dots, 4\}$  indexes the covariates;  $\Xi_{kp}, \Omega_{kp}^2$  and  $\gamma_{kp}$  are the location, scale, and skewness parameters, respectively. Following [19], we also generate covariates  $Z_{kp}$  as non-linear transformation of  $X_{kp}$  such that  $Z_{k1} = \exp(X_{k1}/2)$ ,  $Z_{k2} = X_{k2}/\{1 + \exp(X_{k1})\} + 10$ ,  $Z_{k3} = (X_{k1}X_{k3}/25 + 0.6)^3$  and  $Z_{k4} = (X_{k2} + X_{k4} + 20)^2$ .
To demonstrate that our proposed MR-  $L_{1}$  estimator can handle covariate mismatch across sites, we consider the setting where there exists covariate mismatch across sites, i.e.  $V_{k} = \{X_{k1},X_{k2}\}$  for the target site and  $X_{k} = \{X_{k1},X_{k2},X_{k3},X_{k4}\}$  for the source sites. In Appendix D.1, we also provide results for the setting where there is no covariate mismatch between sites. Specifically, we generate potential outcomes as
$$
Y _ {k} (a) = 2 1 0 + X _ {k} \beta_ {x} + \varepsilon_ {k} \tag {14}
$$
where  $\beta_{x} = (27.4, 13.7, 0, 0)$  for units in the target site and  $\beta_{x} = (27.4, 13.7, 13.7, 13.7)$  for units in the source sites. Similarly, the treatment is generated as
$$
A _ {k} \sim \operatorname {B e r n o u l l i} \left(\pi_ {k}\right) \quad \pi_ {k} = \operatorname {e x p i t} \left(X _ {k} \alpha_ {x}\right) \tag {15}
$$
where  $\alpha_{x} = (-1,0.5,0,0)$  for units in the target site and  $\alpha_{x} = (-1,0.5, - 0.25, - 0.1)$  for units in the source site. With this data generation scheme, the true TATE is  $\Delta_T = 0$
The AIPW- $L_{1}$  estimator, which requires common models across sites, only uses the shared covariates  $(X_{k1}$  and  $X_{k2})$  to specify outcome and treatment models for all sites. On the other hand, our MR- $L_{1}$  estimator allows for different covariates in different sites, so we utilize both shared covariates with the target site and unique covariates to specify the outcome and treatment models in the source sites.
For the site-specific estimators based on MR-  $L_{1}$ , we adaptively mix two outcome models and two treatment models. In particular, we specify the first model with the covariates  $X_{kp}$ , and the second model with the covariates  $Z_{kp}$ .
Table 1: Mean absolute error (MAE), root mean squared error (RMSE), coverage (Cov.), and length (Len.) of  $95\%$  CIs based on 500 simulated data sets in covariate mismatch settings.
Target	SS	IVW	AIPW-L1	MR-L1
MAE	0.108	4.331	0.150	0.107	0.053
RMSE	0.136	4.401	0.186	0.134	0.067
Cov.	0.946	1.000	0.882	0.950	0.944
Len.	0.538	26.024	0.553	0.536	0.253
In Table 1, we observe that the AIPW- $L_{1}$  estimator exhibits similar MAE, RMSE, coverage, and length of confidence intervals as the Target estimator while outperforming the SS and IVW estimators. This is because relying solely on shared covariates leads to significant biases in all source sites (Figure 2, left panel), and the AIPW- $L_{1}$  estimator assigns nearly all of the ensemble weight to the target site so as to reduce bias.
In contrast, the MR- $L_{1}$  estimator outperforms the AIPW- $L_{1}$  estimator by exhibiting substantially smaller MAE, lower RMSE, and better coverage. This improvement can be attributed to the inclusion of unique covariates from the source sites, which allows for the recovery of true models in those sites and contributes to accurate estimation of  $\Delta_T$  (Figure 2, right panel). These findings suggest that neglecting covariate mismatch by solely relying on shared covariates can lead to highly biased results.
7 Case Study
We employ our approach to investigate the treatment effect of percutaneous coronary intervention (PCI) on the duration of hospitalization for individuals experiencing acute myocardial infarction
Figure 2: Estimates of the TATE based on 500 simulated data sets with covariate mismatch comparing the site-specific estimators with nuisance functions estimated by AIPW and by multiply robust model-mixing.
(AMI), one of the leading causes of hospitalization and mortality in the United States. We utilize a dataset from the Centers for Medicare & Medicaid Services (CMS), comprising a representative cohort of Medicare beneficiaries admitted to short-term acute-care hospitals. We select Maine as our target state of interest and incorporate data from the other 48 continental states to augment the Maine-specific treatment effect estimation.
We choose Maine as our target state for studying PCI treatment quality due to its limited AMI patient population. This scarcity of data in Maine necessitates the integration of information from other states for more precise treatment quality assessment. We consider a scenario where we only possess basic patient demographic information, including age, race, gender, and principal diagnosis categories in Maine. In contrast, the other source states have access to more extensive patient medical histories and have collected additional covariates, namely patient comorbidities.
The AIPW- $L_{1}$  estimator requires a homogeneous model across states, so we define outcome and treatment models within each state using common patient demographic variables. By contrast, for our proposed MR- $L_{1}$  estimator, we use patient demographic variables for outcome and treatment models in Maine, while in other states, we use both patient demographic and comorbidity variables for model specification. We adaptively combine two outcome models and two treatment models in the source states. The first model is defined using patient demographic variables, and the second model is defined with both patient demographic and comorbidity variables.
The estimation results of the case study are illustrated in Figure 3 using the five estimators. We observe that the AIPW- $L_{1}$  and our proposed MR- $L_{1}$  estimators yield point estimates that are close to the estimate by the Target estimator, while SS and IVW estimators exhibit substantial bias. The estimated treatment effect of PCI is approximately -7.6 days, indicating that if patients diagnosed with AMI in Maine had received PCI treatment instead of medical management alone, they would have experienced an estimated reduction in their hospitalization duration of approximately 7.6 days $^{2}$ . Both the AIPW- $L_{1}$  and our proposed MR- $L_{1}$  estimators exhibit efficiency gain with smaller standard errors compared to the Target estimator.
Figure 3: Estimates of PCI treatment effect in Maine with covariate mismatch in patient comorbidities
2We assigned patients who died in the hospital a length of stay equivalent to the maximum of either their actual length of stay or the 99th percentile value of the length of stay in the dataset [1]. This modification is intended to prevent hospitals where patients often die early in their hospitalization from inaccurately appearing more efficient than other hospitals. Using the exact length of stay as the outcome may result in a lower estimated effect of the PCI treatment.
To highlight the ensemble weights corresponding to the source states, we generate Figure 4 illustrating the estimated weights associated with each state. States with positive weights are colored in blue, with darker shades indicating larger weights. States with zero weights are uncolored (white). We observe that both SS and IVW estimators integrate all state-specific estimates, regardless of how dissimilar these estimates are to the target estimate, thereby introducing negative transfer. In contrast, with the proposed adaptive ensemble approach, both AIPW- $L_{1}$  and MR- $L_{1}$  estimators only incorporate estimates from a few states whose state-specific estimates are close to the target state estimate, ensuring that the federated estimate closely aligns with the target estimate while achieving an efficiency gain.
Figure 4: Federation weights across states for the PCI treatment effect in Maine using four federated estimators.
We have proposed a novel federated approach for privacy-preserving, multiply robust, and covariate flexible estimation of causal effects. Compared to existing federated methods, our proposed approach accommodates covariate shift and covariate mismatch across sites, while guaranteeing efficient estimation and preserving privacy in the sense that only covariate means of the target samples are shared in a single round of communication. Our proposal allows investigators in each site to have greater flexibility in specifying candidate models by utilizing site-specific information. Moreover, our method utilizes adaptive ensemble weights to avoid negative transfer in the federation process.
In practical scenarios, the proposed method would be particularly valuable in multi-source research settings (e.g., data consortia) lacking a central data collection point. These methods would be beneficial for combining dissimilar data sources from various hospitals. Additionally, our proposed methodology would find utility in other contexts, such as multi-source clinics, firms, or schools needing to assess their performance relative to peer institutions. These situations may involve data privacy or propriety concerns that prevent the sharing of detailed, unit-level data.
To handle high-dimensional covariates, future research can explore ways to jointly model the propensity score and density ratio to reduce the dimension of parameters for population balancing.
In real-world scenarios, a potential weakness of our proposed approach is the risk of breaching patient privacy. We do not claim or test for formal privacy guarantees of the method; our work is privacy-preserving in the sense that only summary-level information is shared between target and source sites. Any given site is not allowed to specify a specific group and query this group's statistics in any other source site's dataset. Consequently, the target site cannot easily perform membership inference attacks. Nonetheless, for a more formal demonstration of privacy preservation, future research could employ the differential privacy (DP) method [11].
We would like to thank Yige Li, Bijan Niknam, Zhenghao Zeng, Wei Li, feedback received at the 2023 American Causal Inference Conference (ACIC) and the 2023 Institute for Operations Research and the Management Sciences (INFORMS) Annual Meeting, and the five anonymous reviewers for their helpful comments and input that greatly enhanced the paper.
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A Equivalence of density ratio weighting and inverse of selection probability weighting
We show that the inverse probability of selection weighting (IPSW) to the site  $k$ ,  $\rho_{k}(V_{i}) = \frac{1 - P(R_{i} = k|V_{i} = v)}{P(R_{i} = k|V_{i} = v)}$  is equivalent to the density ratio weighting  $\zeta_k(V_i) = \frac{P(V_i = v|R_i = T)}{P(V_i = v|R_i = k)}$ .
Under Assumptions (A1) - (A7), the site-specific estimators based on the IPSW for  $\mu_{a,T}$  is
$$
\begin{array}{l} \widehat {\mu} _ {a, k} = \frac {1}{n _ {T}} \sum_ {i = 1} ^ {n} \left[ \frac {I (A _ {i} = a , R _ {i} = k)}{\widehat {\pi} _ {a , k} (X _ {i})} \widehat {\rho} _ {k} (V _ {i}) \{Y _ {i} - \widehat {m} _ {a, k} (X _ {i}) \} \right] \\ + \frac {1}{n _ {T}} \sum_ {i = 1} ^ {n} \Big [ I (R _ {i} = k) \widehat {\rho} _ {k} (V _ {i}) \Big \{\widehat {m} _ {a, k} (X _ {i}) - \widehat {\tau} _ {a, k} (V _ {i}) \Big \} \Big ] \\ + \frac {1}{n _ {T}} \sum_ {i = 1} ^ {n} I \left(R _ {i} = T\right) \hat {\tau} _ {a, k} \left(V _ {i}\right) \tag {16} \\ \end{array}
$$
where  $\widehat{\rho}_k(V_i) = \frac{1 - \widehat{P}(R_i = k|V_i = v)}{\widehat{P}(R_i = k|V_i = v)}$  is an estimator for  $\rho_k(V_i)$ . Applying Baye's rule, we show that the IPSW is equivalent to the density ratio weighting up to a constant,
$$
\begin{array}{l} \rho_ {k} (V _ {i}) = \frac {1 - P (R _ {i} = k \mid V _ {i} = v)}{P (R _ {i} = k \mid V _ {i} = v)} \\ = \frac {P (R _ {i} = T \mid V _ {i} = v)}{P (R _ {i} = k \mid V _ {i} = v)} \\ = \frac {P \left(V _ {i} = v \mid R _ {i} = T\right) P \left(R _ {i} = T\right)}{P \left(V _ {i} = v \mid R _ {i} = k\right) P \left(R _ {i} = k\right)} \\ = \zeta_ {k} \left(V _ {i}\right) \frac {P \left(R _ {i} = T\right)}{P \left(R _ {i} = k\right)}. \tag {17} \\ \end{array}
$$
We re-write (16) by substituting  $\widehat{\rho}_k(V_i)$  with  $\widehat{\zeta}_k(V_i)$ ,
$$
\begin{array}{l} \widehat {\mu} _ {a, k} = \frac {1}{n _ {T}} \sum_ {i = 1} ^ {n} \left[ \frac {I (A _ {i} = a , R _ {i} = k)}{\widehat {\pi} _ {a , k} (X _ {i})} \widehat {\zeta} _ {k} (V _ {i}) \frac {\widehat {P} (R _ {i} = T)}{\widehat {P} (R _ {i} = k)} \{Y _ {i} - \widehat {m} _ {a, k} (X _ {i}) \} \right] \\ + \frac {1}{n _ {T}} \sum_ {i = 1} ^ {n} \left[ I (R _ {i} = k) \widehat {\zeta} _ {k} (V _ {i}) \frac {\widehat {P} (R _ {i} = T)}{\widehat {P} (R _ {i} = k)} \left\{\widehat {m} _ {a, k} (X _ {i}) - \widehat {\tau} _ {a, k} (V _ {i}) \right\} \right] \\ + \frac {1}{n _ {T}} \sum_ {i = 1} ^ {n} I \left(R _ {i} = T\right) \widehat {\tau} _ {a, k} \left(V _ {i}\right). \tag {18} \\ \end{array}
$$
A reasonable estimator  $\frac{\widehat{P}(R_i = T)}{\widehat{P}(R_i = k)}$  is  $\frac{n_T}{n_k}$ . Therefore, we recover our proposed site-specific estimator for  $\mu_{a,T}$ ,
$$
\begin{array}{l} \widehat {\mu} _ {a, k} = \frac {1}{n _ {k}} \sum_ {i = 1} ^ {n} \left[ \frac {I (A _ {i} = a , R _ {i} = k)}{\widehat {\pi} _ {a , k} (X _ {i})} \widehat {\zeta} _ {k} (V _ {i}) \{Y _ {i} - \widehat {m} _ {a, k} (X _ {i}) \} \right] \\ + \frac {1}{n _ {k}} \sum_ {i = 1} ^ {n} \left[ I (R _ {i} = k) \widehat {\zeta} _ {k} (V _ {i}) \left\{\widehat {m} _ {a, k} (X _ {i}) - \widehat {\tau} _ {a, k} (V _ {i}) \right\} \right] \\ + \frac {1}{n _ {T}} \sum_ {i = 1} ^ {n} I \left(R _ {i} = T\right) \hat {\tau} _ {a, k} \left(V _ {i}\right). \tag {19} \\ \end{array}
$$
B Multiply robust estimation for  $m_{a,k}$
For multiply robust outcome estimation within each site  $k$ , we consider a set of  $L$  candidate models for conditional outcomes  $\{m_{a,k}^l (x):l\in \mathcal{L} = \{1,\dots,L\}\}$ . Let  $\hat{m}_{a,k}^{l}(x)$  be the estimates of  $m_{a,k}^{l}(x)$