Combining Aggregate Data and Individual Patient Data in Model‐Based Meta‐Analysis: An Illustrative Case Study of Tofacitinib in Rheumatoid Arthritis Patients

ABSTRACT

Model‐based meta‐analysis (MBMA) utilizes aggregate data (AD) and allows integration of information from multiple studies, which may provide more statistical power to detect clinically relevant treatment effects than an individual randomized controlled trial alone. Access to individual patient data (IPD) is often limited due to confidentiality; therefore, obtaining IPD associated with published literature data is challenging. Thus, to probe predictive covariates, one must rely on an adequate range of aggregate covariate data, or published stratified results could also be used. With access to IPD, or with access to published stratified results, estimates for predictive covariates could be improved. This work is primarily centered on quantifying the potential benefits of having access to IPD when performing MBMA. This was assessed using a 3‐step approach. Two scenarios were explored: one to compare MBMAs with and without access to IPD, assuming no predictive covariates; and another to compare MBMAs with and without access to IPD, where a specific predictive covariate was known to be influential and was used to stratify IPD accordingly. The performance of the method was evaluated for different ratios of IPD studies versus AD studies. In the scenario where an MBMA with covariate was used, instead, the performance of the method was evaluated for different ratios of covariate stratified AD studies versus AD studies. Overall, the benefit of IPD over AD was not evident in the model without covariates, whereas including stratified IPD led to improved covariate model performance.

Study Highlights

• What is the current knowledge on the topic?

  • ○
    Model‐based meta‐analysis (MBMA) is a powerful tool that integrates aggregate data (AD) from multiple randomized controlled trials (RCTs), allowing for broader inference and improved statistical power compared to individual RCTs. It is commonly used to assess treatment effects across heterogeneous studies, even when individual patient data (IPD) is unavailable. However, the lack of access to IPD or stratified AD limits the ability to explore predictive covariates effectively.
• What question did this study address?

  • ○
    This study investigated the potential benefits of incorporating IPD into MBMA, particularly in the context of identifying and quantifying predictive covariates. It specifically aimed to determine how access to IPD or stratified AD influences model performance.
• What does this study add to our knowledge?

  • ○
    The study provides evidence that access to IPD alone does not significantly enhance MBMA performance in the absence of predictive covariates. However, when predictive covariates are relevant and stratified data are available—whether from IPD or stratified AD—model performance improves.
• How might this change drug discovery, development, and/or therapeutics?

  • ○
    This study highlights the value of stratified covariate data in enhancing model‐based decision‐making during drug development. It encourages the availability of stratified results in published literature, which could facilitate more accurate and informative MBMAs. This, in turn, can lead to better‐informed clinical and strategic decisions, even in the absence of IPD.

1. Introduction

Like in conventional meta‐analysis, model‐based meta‐analysis (MBMA) utilizes aggregate data (AD) and allows integration of information from multiple studies, which may provide more statistical power to detect clinically relevant treatment effects than an individual randomized controlled trial (RCT) alone. That is, a key benefit of meta‐analysis is the ability to improve on the precision of observed relative effects by combining results from multiple RCTs. Network meta‐analysis (NMA) and MBMA take this a step further by also allowing for indirect estimates of relative treatment effects between treatments that have never been compared in a head‐to‐head RCT. Additionally, MBMA incorporates parametric pharmacological models and distinguishes itself from traditional meta‐analysis by infusing pharmacologic plausibility into the statistical rigor. MBMA incorporates pharmacologically inspired cause–effect and dose–response relationships and the time course of treatment effects [1]. The model‐based approach of MBMA also allows RCTs with different populations and/or designs to be included in the analysis, as covariate model parameters can be used to explain how these differences may contribute to heterogeneity in RCT outcomes. The variables that modify relative treatment effects (RCT outcomes) are considered predictive covariates. Predictive covariates are in contrast to prognostic covariates, which can be used to explain differences in disease progression outcomes (or placebo effects) that are independent of treatment [2].

As RCTs (and uncontrolled clinical studies) will often publish aggregate results for all treatment arms in the study, MBMA can be used to probe prognostic covariates (on absolute outcomes) and predictive covariates (on relative treatment effects); however, relative treatment effects are typically the primary interest. Both NMA and MBMA are common approaches to estimate direct and indirect treatment effects from a network of RCTs, and the preferred method for this is typically to focus model parameter estimates on relative treatment effects, and this can still be accomplished by keeping the observed outcome as the dependent variable. To do this, fixed effect (or nuisance) parameters are estimated for a reference treatment in every trial. For MBMA, placebo is commonly used as a global reference treatment, even for trials not including a placebo arm. Estimating nuisance parameters for a global placebo reference is commonly referred to as a non‐parametric placebo approach or an unstructured placebo approach [3, 4, 5]. It is also possible to include a placebo model that incorporates a placebo function with prognostic covariates, but this approach is known to be susceptible to biased relative treatment effect estimates, particularly for drugs with fewer (and/or smaller) trials or when placebo response is highly variable and with no discernible trend, thus the non‐parametric placebo approach is highly recommended when analyzing relative effects [6, 7, 8]. If there is interest in understanding prognostic covariates, a model using only placebo data can be used to explore variables that could potentially impact trial outcomes in the placebo arms. If key interest is on predictive covariates (variables that can predict the expected range of treatment effects compared to placebo), a non‐parametric placebo approach with RCTs can explore these in an unbiased way.

It is also important to consider what is needed to compare relative effects between treatments. Unless every patient is given multiple treatments, the difference in effects for different treatments will rely on different groups of patients with similar characteristics receiving those treatments. This is the purpose of an RCT. In an RCT, patients in all treatment groups are randomized to create groups that are, for all intents and purposes, identical other than the treatment they receive. Aggregate results from patients receiving one treatment are then compared to aggregate results from a group receiving another treatment. These groups can potentially be further stratified to explore predictive covariates, but comparisons will always be made using AD. Access to IPD can allow one to stratify results on a variable of their choosing, but relative treatment effects will not be based directly on IPD. Additionally, prognostic covariates can be explored using both AD and IPD, but AD are needed to properly explore predictive covariates; for example, to compare one population with its counterfactual population (identical population but not receiving the treatment).

MBMA is usually applied to understand the competitive landscape considering publicly available data to answer strategic questions related to a specific compound under development. Access to IPD is often limited due to confidentiality; therefore, obtaining IPD associated with published literature data is challenging [9]. Thus, to probe predictive covariates, one must rely on an adequate range of aggregate covariate data (e.g., percent of males or mean age of the population), or published stratified results could also be used (e.g., separate results for males and females). Without access to stratified data, aggregate covariate data will have a significantly narrower range than covariates in IPD, making predictive covariates difficult to quantify. With access to IPD (and hence the possibility to further stratify data), or with access to published stratified results, estimates for predictive covariates could be improved.

This work is primarily centered on quantifying the potential benefits of having access to IPD when performing MBMA of RCTs. This was assessed using a 3‐step approach within the statistical software R version 4.2.3 and package nlme version 3.1 [10, 11]. To the best of our knowledge, example applications like this are limited in the literature. Two scenarios were explored: one to compare MBMAs with and without access to IPD, assuming no predictive covariates; and another to compare MBMAs with and without access to IPD, where a specific predictive covariate was known to be influential and was used to stratify IPD accordingly. Data from rheumatoid arthritis (RA) patients treated with tofacitinib were used as a case study to illustrate access to IPD and/or AD in the MBMA, focusing on the Disease Activity Score (DAS) efficacy endpoint.

2. Methods

2.1. Case Study Example Data

Tofacitinib is a potent, selective Janus kinase (JAK) inhibitor that preferentially inhibits JAK1 and JAK3 [12]. It was approved for the treatment of moderate to severe active RA in adult patients in 2012 [13] and is indicated for the treatment of moderate to severe active RA in adult patients who have responded inadequately to, or who are intolerant to, one or more disease‐modifying antirheumatic drugs [12]. Studies investigating the effect of tofacitinib in RA patients were selected from the CODEX RA clinical outcomes database [14] to illustrate the proposed approach. This database reports results obtained from a systematic search performed on available literature references with a cutoff of 31^st December 2021. We excluded all studies that used a score other than the DAS as the endpoint or used JAK inhibitors other than tofacitinib and retained 11 placebo‐controlled studies for this analysis. The included studies provided information on the following: (1) multiple doses of tofacitinib, including placebo arms; and (2) mean change in DAS score from baseline as the endpoint.

2.2. AD Model Formulation

Based on the AD available and a prior published analysis [15], the relationship between DAS score change from baseline and dose was assumed to follow a maximum effect (Emax) model that can be expressed as follows:

$$Y_{\mathit{ij}}=e_{0i}+\frac{E\max \times {\text{Dose}}_{\mathit{ij}}}{{\mathrm{ED}}_{50}+{\text{Dose}}_{\mathit{ij}}}+{\epsilon }_{\mathit{ij}}$$ (1)

where $Y_{\mathit{ij}}$ is the mean DAS score difference from baseline at week 12 since the start of treatment in arm $j$ of study $i$, ${\mathrm{eo}}_i$ is the placebo response in each study $i$, ${\text{Dose}}_{\mathit{ij}}$ is the dose in arm $\mathrm{j}$ of study $i$, $E\max$ is the maximum drug effect, and ${\mathrm{ED}}_{50}$ is the dose that results in half of the maximal effect. The term ${\epsilon }_{\mathit{ij}}$ specifies residual error assumed to have a normal distribution with a mean of 0 and variance ${\sigma }^2=\mathrm{var}\left({\epsilon }_{\mathit{ij}}\right)=\frac{{{\mathrm{SD}}^2}_{\mathit{ij}}}{N_{\mathit{ij}}}$, where the standard deviation (SD) represents the between‐subject variability reported in sources and $N_{\mathit{ij}}$ is the sample size in arm $j$ of the study i.

Asian ethnicity was identified as a predictive covariate that was reported to have a positive impact on the American College of Rheumatology criteria (ACR) treatment effect, an endpoint used in evaluating the treatment outcome of patients receiving JAK inhibitors [16]. Upon analyzing the data, we found that including Asian ethnicity as a predictive covariate improved the DAS model performance, as indicated by an improved Akaike information criterion (AIC). Thus, the predictive covariate of interest was race (Asian/non‐Asian), and its impact on DAS score changes from baseline was investigated. The race effect was considered a continuous variable that indicated the percentage of Asian individuals, which was similar to a previous NMA study [16]. In particular, the AD covariate effect in the Emax model can be described as an exponential term as follows:

$$f\left(X_{\mathit{ij}},E_{\text{race}}\right)={e^{E_{\text{race}}*X}}_{\mathit{ij}}$$ (2)

$$Y_{\mathit{ij}}=e_{0\mathrm{i}}+\frac{E\max *f\left(X_{\mathit{ij}},E_{\text{race}}\right)*{\text{Dose}}_{\mathit{ij}}}{{\mathrm{ED}}_{50}+{\text{Dose}}_{\mathit{ij}}}+{\epsilon }_{\mathit{ij}}$$ (3)

where $f\left(X_{\mathit{ij}},E_{\text{race}}\right)$ captures the effect of a covariate $X_{\mathit{ij}}$ in arm $j$ of study $i$ and associated parameter $E_{\text{race}}$, $X_{\mathit{ij}}$ represents the % of Asian individuals and $E_{\text{race}}$ is the slope associated with that relationship.

The models illustrated were used as a basis to describe the change in DAS score from baseline dose–response relationship. In particular, Equation (1) was applied to the first type of analysis that uses the model without predictive covariates and combines IPD and AD, whereas Equations (2) and (3) were applied to the second analysis that uses the model with a predictive covariate and relies on AD and stratified AD only. When Equation (1) is applied to IPD instead of AD, the distinction lies in the level at which the error term is defined. For IPD, the error term (${\epsilon }_{\mathit{ijk}}$) is specified at the patient level, i.e., for patient k in study i and arm j, whereas in AD, the error term (${\epsilon }_{\mathit{ij}}$) is aggregated at the study‐arm level.

The difference when applying Equation (2) to stratified AD instead of AD, is that both the error term and the covariate % of Asian will be defined for the two strata s for each study i and arm j (${\epsilon }_{\mathit{ijs}}\text{and}{X}_{\mathit{ijs}}$) and the covariate will be defined as 0 or 100% Asian. The covariate effect, $E_{\text{race}}$, which typically represents the slope of the relationship between the covariate and the outcome, also becomes a relative effect comparing the two strata—Asian versus non‐Asian.

The general workflow of the analysis is illustrated in Figure 1 and includes (1) data simulation and (2) a description of how to combine IPD and AD in a common MBMA (relevant only for the first type of analysis). The steps illustrated in Figure 1 are described below.

FIGURE 1.

General workflow showing the 3‐step approach for combining IPD and AD in a MBMA.

2.3. Data Simulation (Step 0)

We evaluated the base E _max dose–response model in Equations ((1), (2), (3)) based on the AD from 11 studies. Since no IPD were available, a model‐based simulation approach using the identified model was adopted, assuming that the data in each study arm followed a normal distribution. 11 IPD datasets were simulated corresponding to each of the 11 studies, maintaining study‐specific dose levels, number of subjects, and proportion of Asian patients.

For IPD simulation based on the covariate model, the number of subjects in the Asian/non‐Asian strata in each study arm was determined based on the reported Asian percentage and total number of subjects. Practically, the covariate value was set to 100% if the simulated individual was Asian and to 0% if the individual was non‐Asian.

While race is a prognostic factor in RA [17], the race effect was applied only to tofacitinib arms; placebo arms were not stratified by race. Thus, when simulating the placebo group for both Asian and non‐Asian strata using the model with race effect (Equation 3), no difference in DAS change from baseline was applied between the groups. However, in the tofacitinib arms in a real‐world scenario, the Asian strata required a different DAS change from baseline than the non‐Asian strata. The race effect estimated from the full and unstratified AD model was applied to the tofacitinib IPD, while the placebo IPD did not include a race effect.

Based on the simulated IPD, the corresponding AD, which was composed of mean values and associated SDs, was computed. This AD could be used in combination with the simulated IPD for evaluating the use of the IPD/AD combination. In the MBMA example with covariates, IPD were summarized according to their strata (Asian/Non‐Asian) and relative stratified AD was derived for each study arm and sub‐group.

The parameters used for data simulation were defined as “reference values” and are the parameter estimates obtained from step 0 when fitting the models on AD only.

2.4. Combination of IPD and AD in a Common Model (Step 1/2/3)

The method for combining IPD and AD in a common MBMA model was illustrated using a 3‐step approach.

In the first step, we fitted the Emax model presented in Equation (1) to the IPD only and estimated the residual error variance from this model, as is typically done in population analysis with IPD analysis. The residual error terms for patient $k$ of study $i$ and arm j (${\epsilon }_{\mathit{ijk}}$) were assumed to be normally distributed such that

$${\epsilon }_{\mathit{ijk}}~\mathcal{N}\left(0,{\sigma }_{\mathrm{IPD},\mathit{ij}}^2\right)$$

where ${\sigma }_{\mathrm{IPD},\mathit{ij}}^2$ is the residual error variance of the IPD of arm j in study $i.$

In the second step, an MBMA based on the AD studies was developed with the variance of the residual error terms fixed to $\frac{{{\mathrm{SD}}^2}_{\mathit{ij}}}{N_{\mathit{ij}}}$, using the reported SDs of study $i$ and arm $j$. This model reflected the measures of precision based on reported data; hence, no further residual error variance was estimated from this AD‐MBMA. This second step establishes the AD‐based MBMA with fixed residual error variance to ensure correct weighting and model specification for the subsequent integration with IPD in the third step.

In the third and final step, we combined the IPD and AD studies and fitted an integrated model in which the residual error variance associated with IPD was fixed to the value of ${\sigma }_{\mathrm{IPD},\mathit{ij}}^2$ estimated in the first step. The combination of IPD stratified and AD studies was tested using various ratios: 0:11, 1:10, 2:9, 3:8, 4:7, and 5:6, along with all possible permutations of these ratios (1, 11, 55, 165, 330, and 462 permutations for each ratio, respectively).

The residual error variance for the AD was fixed to the reported values of precision in a similar way as in the second step. Standard errors (SEs; or $\sigma$) associated with each data point in the third step were expected to match IPD with the SEs from the first step and with AD with those from step 2, hence validating the integrated model as appropriately combining IPD and AD in a single model.

$${\sigma }_{\mathit{ij}}^2=\left[\begin{array}{c}{\sigma }_{\mathrm{IPD},\mathit{ij}}^2\text{for}\mathrm{arm}j\text{in study}i\text{estimated from}\mathrm{IPD}\\ {\mathrm{\sigma }}_{\mathrm{AD},\mathit{ij}}^2\text{for}\mathrm{arm}j\text{in study}i\text{reported in}\mathrm{AD}\end{array}\right.$$

The impact of using a different ratio of IPD studies versus AD only studies and a different ratio of covariate stratified AD studies versus AD only studies was evaluated by assessing how varying the number of IPD/covariate stratified AD studies would affect parameter uncertainty and how the parameter estimates would deviate from the reference values obtained in step 0 (Figure 1). The optimal combination of IPD/covariate stratified AD and AD studies is expected to provide a lower relative standard error (RSE%) and a closer to 100% ratio of parameter estimates from step 3 of the combined method and reference values identified in step 0 (parameter estimation ratio), defined as follows:

$$\begin{array}{c}\text{Parameter estimation ratio}\left(\%\right)=\hfill \\ \frac{\text{Estimated parameter value in step}3}{\text{Reference value identified in step}0}\times 100\%\hfill \end{array}$$

3. Results

Eleven studies investigating the effect of tofacitinib in patients with RA were selected from the CODEX RA clinical outcomes database. A summary of the data is presented in Table 1 and Figure 2A. Each study comprised a placebo arm and 1–5 dose levels: 2, 6, 10, 20, and 30 mg/day, with the number of patients across dose levels ranging from 14 to 293. The primary outcome of interest was the mean DAS change from baseline at the primary time point, which was uniformly measured at week 12 after treatment initiation across all study arms. The percentage of Asian patients in each study arm was considered a covariate of interest and ranged from 0% to 100%: some study arms had no Asian subjects, some had only Asian subjects, and some had a mixed population (Figure 2B).

TABLE 1.

Summary of reported aggregate data used for the MBMA.

Study ID	Dose level (mg/day)	Number of patients	Asian/Nonasian majority	% Asian
NCT00413660 [18]	0, 2, 6, 10, 20, 30	60, 61, 55, 60, 63, 62	Non‐Asian	0.0, 0.0, 0.0, 1.5, 0.0, 0.0
NCT00550446 [19]	0, 2, 6, 10, 20, 30	59, 54, 51, 49, 61, 57	Non‐Asian	10.2, 9.3, 9.8, 12.2, 8.2, 7.0
NCT00603512 [20]	0, 2, 6, 10, 20	24, 26, 24, 24, 21	Asian	100
NCT00687193 [21]	0, 2, 6, 10, 20, 30	48, 51, 49, 50, 49, 52	Asian	100
NCT00976599 [22]	0, 20	14, 15	Non‐Asian	6.7, 0.0
NCT01484561 [23]	0, 20	44, 93	Non‐Asian	2.0, 3.1
NCT00960440 [24]	0, 10, 20	118, 118, 124	Non‐Asian	8.3, 6.0, 6.0
NCT00847613 [25]	0, 10, 20	160, 321, 316	Non‐Asian	40.8, 42.7, 45.0
NCT00853385 [26]	0, 10, 20	87, 170, 166	Non‐Asian	15.2, 13.9, 18.5
NCT00814307 [27]	0, 10, 20	108, 236, 228	Non‐Asian	12.3, 16.9, 13.1
NCT00856544 [28]	0, 10, 20	147, 293, 291	Non‐Asian	32.1, 35.9, 34.9

FIGURE 2.

Observed DAS score placebo corrected (mean change from control group) stratified by study (A) or by proportion of Asian (B).

The Emax model fits well with the observed AD in the clinical outcome database on the tofacitinib dose and DAS difference from baseline (Figure 3A). The model using % Asian race as a predictive covariate showed better performance (lower AIC, p < 0.01). From both models, individual data were simulated (Figure 3). Figure 3A presents the reported AD with the fitted curve of the AD model (step 0 of Figure 1). Figure 3B represents the simulated data and summarized simulated AD from the Emax model. Figure 3C represents the reported stratified AD with respect to the predictive covariate (% of Asian). Figure 3D represents the simulated data stratified by Asian/non‐Asian and the associated summarized simulated AD from the Emax model.

FIGURE 3.

AD was derived from the IPD simulated data obtained from an Emax model fitted to the reported data of 11 studies. (A) AD derived data using a model without covariates; (B) individual simulated data using a model without covariates; (C) AD derived data color coded with covariates (Asian/non‐Asian) using a model with covariates; and (D) individual simulated data color coded with covariates (Asian/non‐Asian) and stratified aggregated data using a model with covariates. AD, aggregate data; IPD, individual patient data. The solid line represented the Emax model fitting. The error bar represents the standard error. In panel A and C, the solid lines represent the fitted curve from Emax model.

The results (data not shown) indicated that the estimated variance components associated with IPD and AD in the composite model in which both types of data were combined (Step 3 of Figure 1) were the same as the variance components when IPD and AD were analyzed separately in Steps 1 and 2 of Figure 1, respectively. This demonstrates that the combined model could reflect the appropriate weighting associated with each of these two different levels of information. The R package nlme was successfully applied to combine IPD and AD in one common model.

Figure 4 shows the parameter estimate ratios (Figure 4A) and the RSE (Figure 4B) of E _max and ED₅₀ calculated from different IPD/AD ratios using the base model. Implementing IPD in a base MBMA (without covariate) did not change the parameter estimates or their precision; hence there was no advantage in including IPD in the base MBMA. Notably, the ED50 ratios are consistently lower than the Emax ratios, likely due to the dose range in the included studies. Since most trials included doses near or above the ED50, Emax is well‐estimated, whereas ED50 is more sensitive to the distribution of intermediate doses. Limited coverage of doses near the ED50 results in slightly lower parameter estimate ratios.

FIGURE 4.

Parameter estimation of the model combining IPD and AD in the case of no covariate effect. Whiskers represent the standard deviation across multiple simulations.

Figure 5 shows the parameter estimation ratio and the RSE of E _max, ED₅₀, and E _race calculated from different covariate stratified AD/AD ratios. We observed that increasing the proportion of stratified information resulted in noticeably improved parameter estimation precision and accuracy for the Erace parameter, while improvements for Emax and ED50 were more modest (Figure 5).

FIGURE 5.

Parameter estimation of the model combining IPD and AD status using a stratification approach. Whiskers represent the SD across multiple simulations. No whiskers are shown for the 0:11 and 9:2 scenarios (dark blue and yellow bars), as these represent single permutations with no replicates from which to calculate variability.

4. Discussion

The implementation of combined IPD and AD for an MBMA is not common in practice due to challenges related to model assumptions and increased complexity in model development [29]. Internal IPD could be summarized at the study arm level to be leveraged with external AD in an MBMA via a two‐step approach [29, 30]. This can be particularly useful for supporting end‐of‐phase II decisions [29]. There are also other more sophisticated ways to combine IPD and AD in the MBMA population via hierarchical modeling approaches, with different models used for IPD and AD and approximating or imputing the likelihood of AD based on the IPD model [30]. These approaches, however, are subject to critical assumptions, are complex, and oftentimes need to be implemented with challenging computational problems. Most of the publications on IPD and AD combinations in meta‐analysis or meta‐regression are processed in the Statistical Analysis Software (SAS) [31, 32], which is commercial software. On the other hand, R is a comprehensive, open‐source, platform‐independent, freely available programming language that has a massive, worldwide user and contributor base. Our approach successfully combined IPD and AD in a common model that doesn't include covariates using the nlme package in R and was confirmed to be robust in terms of the residual error returned. In our case study, we fit models to both summary‐level and individual patient‐level data to assess the potential advantages of including IPD in the MBMA. When not including individual covariate information, IPD and AD patients should carry similar amounts of information; thus, no significant differences were observed in the modeling results confirming the aforementioned hypothesis.

When performing a meta‐analysis and MBMA, it may become apparent that there is more variation (heterogeneity) between the results of the studies than one might expect by chance. When such heterogeneity is observed, it is important to explore why it exists [33]. Heterogeneity between study results may occur if there are differences in the characteristics of the patients included in each of the studies, and these characteristics are related to treatment efficacy [34]. This heterogeneity could be addressed by including study‐level predictive covariates in the model. Our study showed that covariate‐stratified AD, where IPD were summarized based on covariate stratification, improved the parameter estimation with respect to simply using the AD covariate information.

Another complexity is that data stratification requires selecting appropriate stratification criteria. The choice of covariates for stratification can be informed by subject matter expertise or may result from prior analyses that incorporated non‐stratified covariate information. However, MBMA adds significant value by quantifying the impact of these covariates, thus providing a more objective and data‐driven basis for stratification decisions.

Ecological bias could appear when AD‐based MBMA is used to infer patient‐level properties or when IPD are used to infer group‐level characteristics [35]. According to French et al. [36], the bias will increase in the models where the covariate‐effect relationship enters the model nonlinearly. In our analysis, ecological bias was minimized by carefully matching covariate information across IPD and stratified AD. Note that ecological bias in our base model was not a concern as we simulated for both IPD and AD with the same model and no covariate was present, whereas in our covariate model, ecological bias was not a concern as we decided not to include IPD but rather covariate‐stratified AD. This choice was made because, in the IPD model, the placebo term does not incorporate counterfactual values of covariates. As a result, ecological bias primarily affects prognostic covariates rather than predictive ones.

A limitation of this study is that simulated data were used, which may not fully reflect realistic conditions. A validation using real data needs to be performed to confirm the practicality of the method. This method could be further extended to cases of models with multiple continuous or categorical covariates or with longitudinal MBMA, including random effects. The additional benefit of combining IPD and AD may be evaluated further through simulations by tuning different parameters, such as the number of subjects in the IPD study and the magnitude of between‐subject variability.

Author Contributions

All authors wrote the manuscript, designed the research, performed the research, and analyzed the data.

Funding

The authors have nothing to report.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Data S1: Supporting Information

Pham T.‐N., Largajolli A., Sardu M. L., Maringwa J., Zierhut M. L., and Cheung S. Y. A., “Combining Aggregate Data and Individual Patient Data in Model‐Based Meta‐Analysis: An Illustrative Case Study of Tofacitinib in Rheumatoid Arthritis Patients,” CPT: Pharmacometrics & Systems Pharmacology 15, no. 2 (2026): e70159, 10.1002/psp4.70159.