3 PDA Tasks, Models and Examples
[This section is for readers with some statistical background]
3.1 PDA tasks and models
To date (Sep 02, 2025), the PDA framework includes more than 30 models for federated statistical/machine learning analyses. These models encompass 6 applicational Tasks, namely Regression analysis, Survival analysis, Trial emulation, Causal inference, Experimental design, and Clustering. The users will need to decide the appropriate PDA model based on their applicational task and study setting. Figure 5 is a catalog of all PDA models grouped by Tasks and study setting. Below we introduce the applicational Tasks with example studies, data format, statistical model, and the appropriate PDA model for federated learning. In Section 4, we will provide demo examples with codes for these example studies. The technical details of the models are elaborated in the Appendix.
Figure 5. Tree diagram guiding the choice of PDA model based on the applicational Task and study settings.
3.1.1 Regression
Statistical regression models are widely used in the epidemiological and clinical studies to analyze the association between outcomes and exposure/covariate variables. Depending on the types of the outcome, different regression models may be used. See Table 2 for the list of common outcome types and the regression models. The time-to-event outcome will be introduced in the next subsection of survival analysis.
Application example #1 Opioid use disorder and risk factors A collaborative study was conducted with five participating sites of OneFlorida to study the association between opioid use disorder (OUD) and several relevant clinical risk factors among individuals receiving at least one opioid prescription.
Each participating site extracted EHR records between 01/01/2012 and 03/01/2019 for patients who had opioid prescription (including Codeine, Fentanyl, Hydromorphone, Meperidine, Methadone, Morphine, Oxycodone, Tramadol, Hydrocodone, Buprenorphine), and no cancer or diagnosis of opioid use disorder before their first prescription. Among these patients who were exposed to opioid, a case of OUD is defined as having first diagnosis of opioid use disorder within 12 months after their first prescription and a control is defined as having no diagnosis of opioid use disorder in the entire time window.
Logistic regression was used to model the outcome variable and its association with following risk factors: age, race, gender, and insurance type, Race/ethnicity, and having at least one diagnosis of anxiety, alcohol use disorders, depression, sleep disorders, rheumatoid arthritis, other pain conditions, cannabis-related disorders, nicotine-related disorders, other psychoactive disorders, cocaine-related disorders. The One-shot Distributed Algorithm for Logistic regression (ODAL) model in the PDA toolbox is used as a federated learning approach for this application. The meta-estimator, which takes inverse-variance weighted average of the individual estimates from each site, is used as a naïve benchmark federated learning approach. The estimated odds ratios are shown in the following figure. Compared to the benchmark meta-estimator, the ODAL estimators are very close to the pooled estimators.
Figure X.
Application example #2 International COVID-19 hospitalization length of stay
We investigate the association of COVID-19 hospitalization LOS with patient characteristics, using the EHR and medical claims data from 11 data sources (i.e. “sites”) within and outside of the United States. The data sources include
- The UHG Clinical Discovery Portal data (UHG),
- The OneFlorida data,
- The Stanford Medicine Research Data Repository (STARR),
- Columbia University Irving Medical Center Data Warehouse (CUIMC),
- IBM MarketScan Commercial Database (CCAE),
- IBM MarketScan Medicare Supplemental Database (MDCR),
- Optum de-identified Electronic Health Record Dataset (Optum EHR),
- The Optum COVID data,
- Tufts Medical Center Research Data Warehouse (TRDW),
- The Information System for Research in Primary Care (SIDIAP) from Spain,
- Health Insurance and Review Assessment COVID data (HIRA) from South Korea,
The total number of patients is N=120,609. The patient demographic characteristics include age, gender, and race, and the clinical characteristics include a history of cancer, chronic obstructive pulmonary disease (COPD), heart disease, hypertension, hyperlipidemia, diabetes, kidney disease, and obesity. CCI score is also included as a measure of the overall patient’s health status; the higher the score, the worse the health status is. We also included the admission date (categorized as Q1, Q2, or Q3, i.e., admission in the first, second, or third quarter of 2020, respectively). Race is excluded from the covariates, since race information is missing from some data sources.
We decided to use the linear mixed-effects model (LMM) due to the evident heterogeneity between the data sources (e.g. the South Korea patients have longer LOS than the other countries, despite adjusting for the patient characteristics). The LMM allows the baseline LOS, and the effects of each covariates to vary across sites. The Distributed Linear Mixed-effects Model (DLMM) in the PDA toolbox is used as a federated learning approach for this application. DLMM is mathematically proven to produce identical results compared to the pooled analysis.
Below are the risk factors with significant fixed effects (p-value< 0.001),
- Age and gender are significant risk factors for increased LOS. Age [65, 80) is associated with 0.96 days (95% CI = 0.44–1.48) longer LOS, compared to age [18, 65); and male gender is associated with 0.58 days (95% CI = 0.24–0.92) longer LOS, compared to female gender.
- CCI is a significant risk factor for LOS. CCI [2,5) and 5+ are associated with 1.42 days (95% CI = 0.58–2.27) and 2.68 days (95% CI = 1.16–4.21) longer LOS respectively, compared to CCI [0, 2).
- Hospitalizations in the later time intervals are associated with shorter LOS. Hospitalizations in the second and third quarter are associated with 3.79 (95% CI = 3.03–4.55) and 6.26 (95% CI = 5.23–7.30) days shorter LOS, compared to the first quarter.
- Among those comorbidity conditions, obesity is a significant risk factor of LOS, associates with 0.37 (95% CI = 0.21–0.52) days longer LOS.
Regarding the site-specific random effects, below are comparisons among sites. Compared to other sites,
- The baseline LOS (i.e., the intercept) is longer in in HIRA COVID, and shorter in SIDIAP.
- Age 80+ is associated with shorter LOS in CUIMC, and age 65-80 and male gender are associated with longer LOS in CCAE.
- Higher CCI score is associated with longer LOS in CUIMC.
- CUIMC and Optum EHR have larger effects of admission in the second quarter (i.e., longer LOS compared to the first quarter), and UHG.NE has smaller effects of admission in the second and third quarter.
- History of COPD, kidney and heart diseases are associated with longer LOS in Optum COVID, and history of hypertension, heart disease and diabetes are associated with shorter LOS in CUIMC.
Figure X. The estimated site-specific effects and 95% prediction intervals in the COVID-19 hospitalization LOS study. The site-specific effect is defined as fixed effect plus random effect, and a 95% prediction interval is point estimate ±1.96 * standard error estimate.
3.1.2 Survival analysis
Time-to-event analysis in the epidemiological study, outcome is Time-to-event with possible censoring
[Application example #3] Survival analysis TBA
3.1.3 Trial emulation
Target trial emulation (TTE) is a framework of using observational RWD (e.g. EHR) to mimic the design of randomized controlled trials in clinical effectiveness research.
[Application example #4] TBA
3.1.4 Causal inference
Causal inference is to estimate the treatment effects using the observational RWD.
[Application example #5] TBA
3.2 PDA algorithm workflow
A typical workflow of PDA algorithm is presented in Figure 4:
- the lead site (local site) sends an initial estimate to the collaborating sites,
- the collaborating sites then calculate the intermediate AD and send back to the lead site, and
- the lead site conduct final analysis using all the AD.
The above workflow may vary depending on the PDA model and will be elaborated in each model in Section 4.
Figure 4. A typical workflow of PDA algorithm.
3.2.1 Choice of the lead site
A PDA algorithm relies on a lead site to coordinate the progression of the analytical steps. In some models (e.g. ODA-X models that based on the surrogate likelihood method), the choice of the lead site may also affect the final result as the IPD of the lead site is used in the model fitting. For these models, we recommend choosing the largest site as the lead site. In other models that are completely decentralized (e.g. DLMM, COLA), the choice of the lead site does not affect the final result as only AD of the lead site is used in the model fitting. For these models, we recommend choosing the site with the most knowledge about PDA as the lead site.
3.2.2 Choice of the initial estimate
When an initial estimate is required to start the algorithm (e.g. ODA-X models), the initial estimate can be the estimate of the lead site (e.g. when the lead site is large), or the meta-estimate (by default). If the meta-estimate is used as the initial estimate, there is an extra initialization step that each site sends their individual estimates and its variances to the lead site.
3.2.3 Synthesize step
For some models (e.g. ODA-X models), an optional extra Synthesize step may be conducted to further improve the final results. This step allows each site to repeat Step (3) using their own IPD data, and then take average of the obtained final estimates. This Synthesize step reduces the impact of the choice of the lead site and may result in a better final estimate, with the cost of another round of data transferring.
3.2.4 Total rounds of data transferring
The total rounds of data transferring may vary depending on the PDA model. For some models (e.g. DLMM) the number of rounds is fixed, while for the others, it depends on the choice of model settings (e.g. initial estimate, and Synthesize step). The PDA methods are designed to be non-iterative, and most of the algorithms are finished within 2 rounds of data transferring.
3.2.5 Heterogeneity
Heterogeneity-awareness is one of the core advantages of the PDA framework. We acknowledge that heterogeneity is a broad concept and in the PDA context, it means the model parameters may vary across sites. This between-site heterogeneity is typically incorporated in the PDA methods by assuming random effects models (e.g. DLMM, DPQL), or nuisance non-parametric models (e.g. ODAC-H). This will be elaborated in each model in Section 4.
3.2.6 Aggregate data
All the PDA methods rely on transferring AD for privacy-preserving. The amount of AD may however, depends on the PDA model setting. In general, if the data is of dimension N*p (e.g. N= number of patients, p=number of covariates+outcome), then the AD is of size p*p. In rare cases (e.g. categorical covariates, and N is small), extra privacy-preserving techniques (e.g. suppression) may be applied to further protect the sites from re-identification. This will be elaborated in each model in Section 4.