Highlights
What is already known?
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• Health technology assessments often rely on indirect comparisons for evaluating treatment efficacy. Population adjustment is typically required to reduce the bias in these comparisons.
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• When the trials used to construct the indirect comparison have small sample size and poorly overlapped population, the most popular population adjustment methods, Matching-adjusted indirect comparisons (MAIC) may produce non-sensible estimates without strong parametric assumptions.
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• Regression-based adjustment like parameteric G-computation overcomes the problem of overlap, but relies heavily on parameteric assumptions
What is new?
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• Inspired by Bayesian survey inference literature, we propose G-MAIC, a regression-based population adjustment method that occupies the middle ground between MAIC and parameteric G-computation.
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• Using a comprehensive simulation study, we demonstrate that G-MAIC largely overcomes the instability problem of MAIC under small sample size and poor overlap.
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• We show that population covariate structure can be important to the performance of population adjustment: joint population covariate distribution with non-linear dependency structure can break the implicit assignment model in MAIC, making it potentially more biased than non-adjusted methods.
Potential impact for RSM readers
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• The proposed G-MAIC presents a robust and flexible alternative to the best parametric counterpart, with the potential to accommodate non-parametric regression models and different weighting approaches.
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• Population covariate structure is another important factor to consider when evaluating the performance of population adjustment methods. Focusing solely on standard parametric distributions with different locations may obscure important limitations of these methods.
1 Introduction
Health technology assessment (HTA) plays a vital role in informing healthcare resource allocation decisions in publicly funded health systems, such as those in the United Kingdom, Canada, and Australia. Through systematic evaluation of the clinical and cost-effectiveness of health technologies, including drugs, devices, and procedures,Reference Sorenson, Drummond and Kristensen 1 , Reference O’Rourke, Oortwijn and Schuller 2 HTA provides evidence-based recommendations to help decision-makers maximise health improvements within limited financial resources.
Randomised controlled trials (RCT) form the bedrock of clinical development and are required to demonstrate safety and efficacy for market approval.Reference Guyatt, Oxman and Akl 3 , Reference Sorenson, Drummond and Kristensen 4 However, for novel interventions, reimbursement decisions are often made without direct RCT evidence on the relevant head-to-head comparisons (e.g., new vs existing drugs, rather than placebo).
In HTA, methods for ‘indirect’ comparisons estimate the relevant incremental clinical benefits even in the absence of RCTs directly comparing the two competing interventions. Standard indirect comparisons mainly focus on ‘connected’ networks, where two alternative interventions are assessed in two RCTs against a common comparator (often placebo or standard of care). In this case, the effect of the two interventions can be obtained by contrasting their relative effects against the common comparator. This is known as ‘Bucher’s method’Reference Bucher, Guyatt, Griffith and Walter 5 and preserves randomisation within trial to provide an unbiased estimate of the treatment effect, under the assumption that it is constant across trials.Reference Dias, Sutton, Ades and Welton 6
Unfortunately, Bucher’s method is biased when effect-modifiers exist and are distributed differentially in the two trials, a common situation in practice. Thus, to provide accurate inference, analysts have to first ‘transport’ the relative treatment effects to a common population and then apply Bucher’s method. In a two-trial network with individual participant data (IPD) available from both trial populations, one could re-weight the IPD from one of the trials by the inverse odds of assignment to that study,Reference Westreich, Edwards, Lesko, Stuart and Cole 7 which can balance the distribution of effect-modifiers across trials. However, due to confidentiality and commercial reasons, IPD are rarely available for all relevant sources. It is more common to have IPD from one company’s own sponsored trial, but only aggregated-level data (ALD), usually from summary statistics reported in published studies, for the comparator’s trial. The limited data accessibility poses additional challenges and has received an increasing amount of attention in recent years.Reference Tremblay, Chandiwana, Dolph, Hearnden, Forsythe and Monaco 8 – Reference Kalf, Dawoud and Bregman 10
When only ALD are available from one study in a two-trial network, population adjustment methods are either classified as ‘weighting-based’ or ‘regression-based.’ The most popular weighting-based method is Matching-Adjusted Indirect Comparisons (MAIC),Reference Signorovitch, Wu and Yu 11 , Reference Signorovitch, Sikirica and Erder 12 while the most common regression-based methods are Simulated Treatment Comparisons (STC)Reference Ishak, Rael and Phatak 13 and parametric G-computation.Reference Remiro-Azócar, Heath and Baio 14 Recently, another method has been specifically designed for evidence synthesis in larger networks and is known as Multi-Level Network Meta Regression (ML-NMR).Reference Phillippo, Dias and Ades 15 As a network evidence synthesis method, ML-NMR estimates a full outcome regression model by jointly synthesising data from multiple IPD and ALD studies. The resulting model naturally accommodates conditional estimands, and can be further marginalised over any chosen target population to produce the marginal estimand of interest—a key advantage for informing population-level decision-making.Reference Phillippo, Dias and Ades 15 – Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton 17
In the broader context of HTA, multiple relevant studies frequently exist beyond the simplified two-trial network we mentioned above, and methods like MAIC and STC are inherently limited to indirect comparisons of two trials. Consequently, such methods cannot coherently synthesise multiple sources of evidence across complex networks due to target population mismatch. Nevertheless, two-trial networks remain highly relevant, particularly when evidence is limited—such as shortly after regulatory approval when a novel treatment is only compared against placebo or standard care in an efficacy trial. Under such circumstances, indirect comparisons based on two-trial networks may offer critical preliminary evidence regarding comparative effectiveness while broader comparative evidence is still developing.
MAIC is a weighting-based method that generates weights based on the odds of assignment to the ALD trial in a limited data context. The weights, i.e., the odds of being assigned to the ALD trial, are not directly estimable due to the lack of IPD, but can be obtained through optimisation if combined with a moment-matching constraint.Reference Signorovitch, Wu and Yu 11 The optimised weights are used to fit a weighted regression model on the available IPD, with a treatment indicator as the only covariate. The estimated coefficient from this model quantifies the marginal treatment effect that might have been observed in the ALD population. MAIC provides unbiased estimate of the indirect treatment effect under the correct specification of the trial assignment model.Reference Remiro-Azócar, Heath and Baio 18 , Reference Wang 19 However, poor overlap of the covariates combined with small sample sizes, a common issue in practice, can lead to extreme weights, potentially leading to higher bias than Bucher’s method.Reference Phillippo, Dias, Ades and Welton 20 The associated bootstrap variance estimatorReference Efron 21 can also be unstable, with weight estimation failing in some bootstrapped datasets and the estimated variance being unrealistically large.Reference Phillippo 22
Regression-based methods can overcome the problem of unstable weights under poor population overlap by extrapolating beyond the covariate space of the IPD. A key consideration in using regression-based methods is the choice between marginal and conditional estimands; the former represents the average treatment effect averaged over a certain population, while the latter is conditional on specific covariate profile. As an example, the Simulated Trial Comparison (STC),Reference Ishak, Proskorovsky and Benedict 23 which ‘reads off’ the regression coefficients as the estimate of population-adjusted treatment effects, typically targets the conditional estimand.Reference Remiro-Azócar, Heath and Baio 18 , Reference Remiro-Azócar 24 This would not align with the marginal estimand for non-collapsible outcome measures, such as odds ratios and hazard ratios.Reference Remiro-Azócar 24 , Reference Daniel, Zhang and Farewell 25 We acknowledge that the selection of an estimand is critical but also nuanced, with profound implications on the transportability of the results, and there are varying perspectives regarding this choice.Reference Remiro-Azócar 24 , Reference Phillippo, Dias, Ades and Welton 26 This manuscript will primarily focus on methods target the same estimand as used in MAIC, i.e., the marginal estimand, setting aside the broader debate for a more focused discussion.
Another regression-based adjustment method, Parametric G-computation, provides a marginal estimand even for non-collapsible outcome measures.Reference Remiro-Azócar 24 , Reference Phillippo, Dias, Ades and Welton 26 , Reference Remiro-Azócar, Heath and Baio 27 It predicts the treatment effect in the ALD population by fitting an outcome regression model using IPD and extrapolating the fitted relationship to the simulated ALD population. However, this method assumes the covariates in the ALD population follow specific parametric distributions. The covariate-outcome relationship must also be correctly specified beyond the boundary of IPD, making analysts cautious about trusting results based on extrapolation and a fully simulated population.
We propose a method that occupies the middle ground between MAIC and parametric G-computation. This regression-based method requires a parametric outcome model but does not parametrise the covariate distribution in the ALD population. Instead, the ALD population is approximated by re-weighting the IPD population using mean-balancing weights, similar to MAIC. To coherently propagate uncertainty, we also adopt a Bayesian framework. The variance of the adjusted treatment effects can be estimated directly from the predicted treatment effects in each draw of approximated ALD population, operationalised via Bayesian bootstrap.Reference Rubin 28 This article introduces the population adjustment problem and discusses two main methods. We then introduce our proposed method before benchmarking it against existing methods in a simulation study. The simulation results are discussed, along with general issues in population adjustment, in the final section.
2 Overview of population adjustment
We aim to estimate the incremental benefit of intervention A compared to intervention B, to inform a decision model. We have evidence from two randomised trials, one for A and one for B, which may or may not have the same comparator to form a connected or disconnected ‘network’ (Figure 1a and 1b, respectively). An anchored network occurs when treatments A and B are connected through a common treatment C, allowing for unbiased indirect comparisons if the relative treatment effect remains constant. Conversely, the unanchored network (Figure 1b) requires stronger assumptions: the A vs B effect is unbiased only if the absolute effect of treatment A is constant, which requires the two trial population to be exchangeable in terms of both effect modifiers and prognostic factors. Given that this requirement can be too stringent to be realistic, we will focus on anchored comparisons.
Figure 1 (a) shows an example of ‘Anchored comparison’, where the two treatments of interest A and B are compared against the common comparator C in two separate trials; (b) shows an example of ‘Unanchored comparison’ where the two treatments of interest A and B are compared against different comparators C and D in respective trials.
For all discussions concerning anchored comparisons, we use the following notation for clarity:
$d_{AB}, d_{AC}, d_{BC}$
refer to the A vs B, A vs C and B vs C treatment effects respectively. The standard ‘Bucher’s method’ estimates
$d_{AB} = d_{AC} - d_{BC}$
. As mentioned, this gives biased estimates in the presence of effect-modifiers. Population adjustment methods are thus required for a ‘fair comparison.’
2.1 MAIC
MAIC is likelihood reweighting applied in the limited data context.Reference Nie and Soon
29
,
Reference Nie, Zhang, Rubin and Chu
30
It determines the estimated treatment effect in the ALD population by reweighting the individual-level likelihood contributions in the IPD according to trial assignment odds, effectively estimating the treatment effect in a pseudo population that is exchangeable with the ALD. The calculation of weights consists of two pillars: a parametric model for the probability that each patient is enrolled in the ALD trial and a constraint that ensures equal means of effect modifiers across both trials. When MAIC was introduced,Reference Signorovitch, Wu and Yu
11
the weights were trial assignment odds, parametrised using their corresponding K effect-modifiers
${\boldsymbol{X}^{\text {EM}}_{\text {IPD}}=\left (X^{\text {EM}}_{1,\text {IPD}},\ldots ,X^{\text {EM}}_{K,\text {IPD}}\right )}$
, on the log scale. In other words, if
$p_i$
is the probability of individual i being assigned to the ALD trial, then the (log)-weights are computed as
$$ \begin{align} \log w_i := \log\left(\frac{p_i}{1-p_i}\right) = \mbox{logit}(p_i) = \alpha_0 + \sum_{k=1}^K \alpha_k X^{\text{EM}}_{i,k,\text{IPD}}. \end{align} $$
When we do not have access to individual-level data for the ALD population, a complete likelihood function cannot be formed and the coefficients in Equation 1 cannot be estimated using standard statistical methods.
Nevertheless, if we combine the expression for the
$w_i$
’s with a mean-balancing constraint:
$$ \begin{align} \frac{\sum_{i=1}^n w_i\mathbf{X}_{i,\text{IPD}}^{\text{EM}}}{\sum_{i=1}^n w_i} = \bar{\mathbf{X}}^{\text{EM}}_{\text{ALD}}, \end{align} $$
where
$\bar {\mathbf {X}}^{\text {EM}}_{\text {ALD}}$
is the mean of the effect-modifiers in the ALD population, Equation (2) can be solved as an optimisation problem. The optimised weights are then used in a weighted regression with treatment indicator as the only covariate, equivalent to applying an unadjusted regression model to a population where the distribution of effect-modifiers are balanced, on average.
To quantify uncertainty in the MAIC estimator, we can use non-parametric bootstrap, which repeatedly samples the IPD to compute the weights and run a weighted regression for each re-sampled dataset. This generates a ‘bootstrapped distribution’ of the MAIC estimates, whose empirical variance can be used as the estimate for variance of the MAIC estimator.
MAIC is appealing as it only assumes a logistic regression model for trial assignment. The correctness of this assignment model rest on the assumptions that the trial assignment odds are log-linear in covariates.Reference Wang 19 , Reference Josey, Berkowitz, Ghosh and Raghavan 31 From a population density estimation perspective, reweighting the IPD by MAIC weights lead to an exponential tilted population distribution. Implicitly, this implies that the assignment model would be mis-specified if the joint distributions of effect-modifiers are complex, or exhibit non-linear dependence structure. Additionally, the mean-balancing requirement in equation (2) enhances robustness; the final population-adjusted estimate will be unbiased irrespective of the trial assignment model, if the outcome model is linear in the covariates on the natural scale.Reference Wang 19 , Reference Josey, Berkowitz, Ghosh and Raghavan 31 However, popular models in HTA analysis, e.g., logistic regressions and survival models, are not linear in the covariates; thus, the accuracy of MAIC requires a correct trial assignment model. Standard MAIC estimates would be biased if the joint distributions of effect-modifiers within the ALD trial populations become complex, an example would be mixing in skewed or heavy-tailed distributions.Reference Jackson, Rhodes and Ouwens 32
2.2 Parametric G-computation
Regression-based methods view population adjustment as a prediction problem. In order to target a marginal treatment effect similar to MAIC, the marginal distribution of the outcome variable must be modelled in the target population with and without treatment.Reference Remiro-Azócar, Heath and Baio 14 Parametric G-computation uses IPD for the AC trial to fit an outcome model, which predicts the outcomes under different treatment conditions and computes the treatment effects of interest in the fully-parametrised ALD trial population.
Although parametric G-computation has no theoretical restrictions on the type of outcome, here we illustrate it for a binary outcome
$Y_i$
, where the probability of occurrence is described for each individual
$i=1,\ldots ,n$
as
$$ \begin{align} \mbox{E}(Y\mid \boldsymbol{X},\boldsymbol{T}) = g\left(\beta_0 + \boldsymbol{\beta_{\text{PV}}}\boldsymbol{X}^{\text{PV}} + \left(\beta_{trt} + \boldsymbol{\beta_{\text{EM}}}\boldsymbol{X}^{\text{EM}}\right)\boldsymbol{T}\right), \end{align} $$
where
$g(\cdot )$
is the inverse logit function and the term
$\mbox {E}(Y \mid \boldsymbol{X},\boldsymbol{T})$
is the probability that the outcome occurs conditional on the treatment assignment
$\boldsymbol{T}$
and a given covariates profile. The model includes: an intercept
$\beta _0$
for the baseline odds; a coefficient
$\beta _{trt}$
for the treatment effect that does not vary according to covariates; a vector
$\boldsymbol{\beta _{\text {PV}}}$
for the effect of the prognostic variables
$\boldsymbol{X}^{\text {PV}}$
; and a vector
$\boldsymbol{\beta _{\text {EM}}}$
for the effect of the effect-modifiers
$\boldsymbol{X}^{\text {EM}}$
.
Remiro-Azocar et al. implemented this outcome regression in a Bayesian framework.Reference Remiro-Azócar
24
After fitting the regression model, parametric G-computation is based on the distribution of effect-modifiers and prognostic variables in a target population. If we set
$\boldsymbol{X}=(\boldsymbol{X}^{\text {PV}},\boldsymbol{X}^{\text {EM}})$
to include both prognostic variables and effect-modifiers, we then define the theoretical covariates distribution in the target population as
$f_{\boldsymbol{X}}(\boldsymbol{x})$
. Conditional on coefficients
$\boldsymbol{\beta }$
, individual-level predictions are aggregated to obtain the distribution of averages of hypothetical outcomes:
$$ \begin{align} \mbox{E}(Y^{pred} \mid \boldsymbol{\beta}) & = \int_{\boldsymbol{x} \in \boldsymbol{S}} g\left(\beta_0 + \boldsymbol{\beta_{\text{PV}}}\boldsymbol{x}^{\text{PV}} + (\beta_{trt} + \boldsymbol{\beta_{\text{EM}}}\boldsymbol{x}^{\text{EM}})T\right) f_{\boldsymbol{X}}(\boldsymbol{x}) d\boldsymbol{x}\nonumber \\ & \approx \int_{\boldsymbol{x} \in \boldsymbol{S^\prime}} g\left(\beta_0 + \boldsymbol{\beta_{\text{PV}}}\boldsymbol{x}^{\text{PV}} + (\beta_{trt} + \boldsymbol{\beta_{\text{EM}}}\boldsymbol{x}^{\text{EM}})T\right) \widehat{f}_{\boldsymbol{X}}(\boldsymbol{x}) d\boldsymbol{x} , \end{align} $$
where
$\boldsymbol{S}$
is the support of the theoretical distribution
$f_{\boldsymbol{X}}(\boldsymbol{x})$
, which is generally unknown in practice. Thus, we approximate it using a finite population approximation
$\widehat {f}_{\boldsymbol{X}}(\boldsymbol{x})$
, defined over its support
$\boldsymbol{S^\prime }$
.
The posterior distribution of the conditional mean under either treatment condition,
$\widehat {\mu }(t)$
, can be found using a simulation approach by incorporating L posterior draws of the regression coefficients
$\boldsymbol{\beta }$
into Equation 4. Thus, the l-th draw of the average hypothetical outcomes can be calculated as
$$ \begin{align} \widehat{\mu}^l(t) = \int_{\boldsymbol{x} \in \boldsymbol{S^\prime}} g\left(\beta_0^l+\boldsymbol{\beta_1}^l\boldsymbol{x}^{\text{PV}} + (\beta_{trt}^l+ \boldsymbol{\beta_{EM}}^l\boldsymbol{x}^{\text{EM}})t \right)\widehat{f}_{\boldsymbol{X}}(\boldsymbol{x})d\boldsymbol{x}, \end{align} $$
where t is the treatment condition of interest.
In standard G-computation,
$\widehat {f}_{\boldsymbol{X}}(\boldsymbol{x})$
is usually the empirical covariate distribution of the sample IPD. However, in population adjustment,
$\widehat {f}_{\boldsymbol{X}}(\boldsymbol{x})$
is based on simulating
$N^*$
covariate profiles from the assumed ALD trial population,
$\boldsymbol{x}_i$
,
$i = 1,\dots , N^*$
. Thus, (5) can be simplified to
$$ \begin{align*} \widehat{\mu}^l(t) = \sum_{i=1}^{N^*} g\left(\beta_0^l+\boldsymbol{\beta_1}^l\boldsymbol{x}^{\text{PV}}_i + (\beta_{trt}^l+ \boldsymbol{\beta_{EM}}^l\boldsymbol{x}^{\text{EM}}_i)t \right). \end{align*} $$
Finally, the lth draw from the posterior of the treatment effect can be computed as
$$ \begin{align*} \widehat{\Delta}^l = \left. \left(\frac{\widehat{\mu}^l(1)}{1-\widehat{\mu}^l(1)}\right) \right/ \left(\frac{\widehat{\mu}^l(0)}{1-\widehat{\mu}^l(0)}\right). \end{align*} $$
By directly parametrising the ALD population and extrapolating the outcome model beyond the covariate space of the IPD trial, parametric G-computation avoids the problem of unstable weights caused by the limited population overlap. In simulation studies, it provides more efficient estimates with minimal bias.Reference Remiro-Azócar, Heath and Baio 14 However, parametric G-computation achieves this at the cost of stronger assumptions; it relies heavily on parametric assumptions—both the parametric representation of the ALD population and the outcome model must be correct for unbiased estimation. While the effect of model misspecification is still under investigation, results based on extrapolation and a fully simulated population may not be taken at face value.
3 G-MAIC
To overcome the limitations of the two procedures shown above, we propose a method, called G-MAIC, that reduces the small-sample bias and improves precision compared to MAIC, without explicitly parametrising the ALD trial population. G-MAIC combines weighting adjustment with G-computation and uses a variance estimator based on Bayesian bootstrap.Reference Rubin 28 As we will demonstrate in the simulation study in Section 4, it offers a promising alternative to MAIC, performing comparably to parametric G-computation.
Table 1 shows the connections between G-MAIC, MAIC, and parametric G-computation. G-MAIC requires a parametric outcome model, similar to parametric G-computation. However, instead of directly parametrising the distribution of effect-modifiers, G-MAIC indirectly parametrises
$\widehat {f}_{\boldsymbol{X}}(\boldsymbol{x})$
in Equation (5) by re-weighing the IPD using MAIC weightsFootnote
1
, such that it resembles a population that is exchangeable to the ALD population. Similar to MAIC, this approach creates a pseudo population where the distribution of effect modifiers is, on average, balanced. The only difference being that G-MAIC reweighs the covariate instead of individual likelihood contribution as in MAIC.
Table 1 General building blocks for population adjustment methods targeting the marginal treatment effects
Note: ‘Implicit’ outcome model refers to the mean-matching step in MAIC guarantees its unbiasedness when the outcome is linear in covariates; ‘Indirect’ covariate model refers to first specifying a trial assignment model for weights estimation, and then use weights for approximating the ALD trial.
In order to propagate uncertainty from the approximated distribution of effect-modifiers in the ALD population to the estimated treatment effect, we propose a variance estimator based on Bayesian bootstrap.Reference Rubin 28 This variance estimator is more stable than the standard bootstrap used in MAIC, which can reduce the covariate space and result in extreme weights in poorly overlapped scenarios. As explained in the next section, the Bayesian bootstrap estimator treats approximating the effect-modifier distribution in the ALD population as a prediction task based on limited data, and considers the predictive distribution of the re-weighed population.
3.1 Covariate approximation using Bayesian bootstrap
In our view, the primary challenge in population adjustment lies in accurately approximating the distribution of effect-modifiers, while adequately quantifying uncertainty. This is because explicit parametric assumptions must be made (which may be hard to justify) and, in addition, the uncertainty in the population approximation is often overlooked, as only a single draw from the assumed parametric distribution is utilised for the analysis.
We attempt to address these issues in the following manner. Consider the following approximation of
$\widehat {f}_{\boldsymbol{X}}(\boldsymbol{x})$
in (5):
$$ \begin{align} \widehat{f}_{\boldsymbol{X}}(\boldsymbol{x}) = \sum_{i =1}^n m_{i}\delta_{\boldsymbol{X}_i}(\boldsymbol{x}) , \end{align} $$
where
$\delta _{\boldsymbol{X}_i}(\boldsymbol{x})$
indicates that the effects modifiers are set to a fixed individual ‘profile’
$\boldsymbol{x}$
, without uncertainty, effectively assuming a degenerate distribution for
$\boldsymbol{X}$
. The values
$\boldsymbol{m} = \{m_1, m_2, \ldots m_n\}$
indicate weight parameters with the additional constraint that
$\sum _{i=1}^n m_i = 1$
. By assigning different weights to different values of the effect modifiers, (6) can be employed to represent the empirical distribution of effect-modifiers in a finite sample. Additionally, if
$\boldsymbol{m}$
is drawn from a probability distribution, the uncertainty in
$\widehat {f}_{\boldsymbol{X}}(\boldsymbol{x})$
can be directly modelled.
Under a Bayesian framework, if the IPD and the ALD population can be seen as exchangeable, we can construct an approximation to
$\widehat {f}_{\boldsymbol{X}}(\boldsymbol{x})$
based on the empirical distribution of effect-modifiers in the IPD trial. Assume
$\boldsymbol{m}$
is drawn from an improper Dirichlet prior
$\boldsymbol{m} \sim \mbox {Dirichlet}(\boldsymbol{0}_n)$
and suppose each observation corresponds to a unique covariate profile. The empirical data can be considered as a realisation of a Multinomial distribution with parameters equal to
$\frac {1}{n}$
for all n observed values. Combining these data with the improper prior yields the conjugate posterior
$\boldsymbol{m}\mid \boldsymbol{X} =\boldsymbol{x}\sim \mbox {Dirichlet}(\boldsymbol{1}_n)$
.
As explained by Oganisian et al.,Reference Oganisian and Roy 33 this is the Bayesian bootstrap developed by Rubin.Reference Rubin 28 More intuitive interpretations of this procedure can be found in Bayesian survey inference under the term ‘Polya Sampling,’Reference Malya and Glen 34 , Reference Lo 35 where the procedure is applied to impute the upsampled population based on sampled units. Bayesian bootstrap can also be conceptualised as ‘predictive resampling’ in Bayesian predictive inference, updating the ‘one-step predictive distribution’ by repeatedly resampling the Bayesian posterior.Reference Fong, Holmes and Walker 36 Both ideas show that, under exchangeability assumptions, Bayesian bootstrap is equivalent to modelling predictive distributions conditional on the known population.
In population adjustment, the two trial populations are not exchangeable. However, the re-weighed IPD based on MAIC weights can be viewed as being exchangeable to the ALD population, to a reasonable degree. By applying Bayesian bootstrap to the re-weighed IPD, we are essentially imputing the ALD population based on the pseudo population, with uncertainty of approximating the ALD contained within the predictive distributions.
3.2 Statistical implementations
As stated in Table 1, G-MAIC assumes a parametric regression model similar to G-computation. In the marginalisation step, however, instead of averaging over a single simulated population, marginalisation is combined with Bayesian bootstrap to compute the average treatment effect with its associated uncertainty.
We have shown that applying Bayesian bootstrap to the re-weighted IPD gives us the posterior predictive distribution of the adjusted population. Therefore, each draw of the population-adjusted average outcomes under either treatment condition can be calculated as a weighted average by combining one posterior draw of the conditional mean with a set of random Dirichlet weights. The proposed population-adjusted treatment effect estimation can be implemented as follows.
We start by computing the normalised MAIC weights
$\boldsymbol{W}$
based on (1) and (2), where
$\sum _{i=1}^n w_i = N^{\text {IPD}}$
, using the full sample. This then serves as the basis to construct our random weight vector
$\boldsymbol{m}$
; specifically, we replace the frequency of 1 in standard Bayesian bootstrap with the pseudo frequency
$\boldsymbol{W}$
. Starting with the same non-informative prior for random weights, by the same Dirichlet-Multinomial conjugacy, the posterior distribution for the random weights
$\boldsymbol{m}$
is:
$\boldsymbol{m} \mid \boldsymbol{W} \sim \mbox {Dirichlet}(\boldsymbol{W})$
, the random Dirichlet weights can then be attached to the observations as in Equation (6).
Thus, the new adjusted distribution can be represented as
$$\begin{align*}\begin{aligned} \boldsymbol{\boldsymbol{m}} \mid \boldsymbol{W} \sim \mbox{Dirichlet}(\boldsymbol{W}) \\ \widehat{f}_{\boldsymbol{X}}(\boldsymbol{x}) = \sum_{i =1}^n m_{i}\delta_{\boldsymbol{X}_i}(\boldsymbol{x}).\\ \end{aligned} \end{align*}$$
After fitting the same outcome regression model as in parametric G-computation, the full posterior inference for the treatment effect proceeds as in standard Bayesian G-computation. For the lth draw from the posterior distribution of
$\{\beta _0^l, \boldsymbol{\beta _1}^l,\beta _{trt}^l,\boldsymbol{\beta _{EM}}^l \}$
, a weight vector
$\boldsymbol{m^l}$
is drawn:
$\boldsymbol{m^l} \mid \boldsymbol{W} \sim \mbox {Dirichlet}(\boldsymbol{W})$
. The causal contrast can be computed as follows:
-
• For both T = {0,1}, average the conditional mean over the specific distribution of effect-modifiers:
$$\begin{align*}\mu^l(t) = \sum_{i=1}^{n} m^l_i \hspace{1mm} g\{\beta_0^l+\boldsymbol{\beta_1}^l\boldsymbol{x}_i + (\beta_{trt}^l+ \boldsymbol{\beta_{EM}}^l\boldsymbol{x}_i)t \}. \end{align*}$$
-
• Compute the causal contrast (e.g., the odds ratio):
$$\begin{align*}\Delta^l = \frac{\mu^l(1)/\left[1-\mu^l(1)\right]}{\mu^l(0)/\left[1-\mu^l(0)\right]}. \end{align*}$$
Repeating this step for
$l = 1,2,\dots , L$
posterior draws gives L draws for the population-adjusted causal odds ratio, which could then be used to compute any summary of interest.
3.3 Hypotheses and remarks
The proposed G-MAIC method employs the same logic as standard MAIC, which aims to estimate the treatment effect in a pseudo population with balanced effect-modifiers. However, the two methods differ in point and variance estimation. Instead of using an unadjusted estimator, G-MAIC utilises a ‘regression-then-marginalisation’ approach, which can increase efficiency, particularly for small sample sizes. The approach, coupled with Bayesian bootstrap that relies on MAIC weights calculated using the full sample, makes G-MAIC more reliable than traditional MAIC, as we will demonstrate in Section 5.
However, G-MAIC is still sensitive to population overlap. Limited population overlap leads to weight concentration, resulting in an underestimation of the variance of treatment effects. This problem becomes more severe as the overlap decreases. Therefore, when population overlap is poor, parametric G-computation is preferred due to its ability to extrapolate. Based on the above features of G-MAIC, we undertook the simulation study to verify the following hypotheses:
-
1. G-MAIC will have smaller variance than MAIC in small samples.
-
2. G-MAIC will have a more stable variance estimator than MAIC.
-
3. G-MAIC will have incorrect nominal coverage when overlap is poor.
-
4. G-MAIC will have worse under-coverage for small effective sample sizes (ESS).
-
5. G-MAIC will provide worse estimates than parametric G-computation with poor overlap.
4 Simulation studies
4.1 Aim
This simulation study aims to examine whether G-MAIC can improve over standard MAIC in challenging anchored two-trial scenarios. We also aim to benchmark the performance of G-MAIC against parametric G-computation, which is not affected by poor trial population overlap. We expect G-MAIC to offer efficiency gains while being relatively robust when the trial assignment model of MAIC is mis-specified. We will evaluate the following metrics under different data-generating mechanisms: (1) unbiasedness; (2) variance unbiasedness; (3) coverage; (4) precision. The simulation study is reported following the ADEMP (Aim, Data-generating mechanisms; Estimands; Methods; performance measures) structure.Reference Morris, White and Crowther 37 The simulation study was implemented in R software version 4.1.1, 38 with R code available on github at https://github.com/garoc371/G-MAIC/tree/main.
4.2 Data-generating mechanisms
We consider a binary outcome, generated under a logit model as in (3). The two trials are
$AC$
and
$BC$
, involving three treatment
$(A,B,C$
) with C being the common comparator.
We first generate IPD for both trials, with the
$AC$
trial designated as the IPD trial. The IPD consist of individual-level outcomes
$\boldsymbol{Y}$
, the treatment assignment status
$\boldsymbol{T}$
and a matrix of five covariates
$\boldsymbol{X}$
. All covariates are assumed to be prognostic variables and effect-modifiers, i.e.,
$\boldsymbol{X_{EM}} = \boldsymbol{X}$
in equation (3). The BC trial is the ALD trial so the simulated IPD are aggregated to obtain covariate summaries; overall event counts and the number of participants by treatment, where, specifically
$N_{B},N_C, n_{B}, n_{C}$
are the sample sizes and overall event count in arm B and C. We fix the size of the ALD trial at 600, corresponding to a typical Phase III trial,Reference Stanley
39
with a
$1:1$
allocation ratio.
The model for the probability of experiencing the outcome is:
$$\begin{align*}\mbox{logit}(\theta_i) = \beta_0 + \boldsymbol{\beta_1}\boldsymbol{X_i^{\text{EM}}} + (\beta_{trt} + \boldsymbol{\beta_{\text{EM}}}\boldsymbol{X_i^{\text{EM}}})T_i, \end{align*}$$
where
$\beta _0$
is the fixed control group event rate. The strength of effect-modification
$\boldsymbol{\beta _{EM}}$
is assumed to be the same under the assumption of shared effect-modifiers,Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton
17
and we set
$\boldsymbol{\beta }_{EM} = -\log (0.6)$
, corresponding to strong effect-modification. The prognostic strength
$\boldsymbol{\beta }_1$
is also assumed to be the same and set to
$\boldsymbol{\beta }_1 = -\log (0.8)$
. The treatment effects are assumed to be equally large in both trials with
$\beta _{AC} = \beta _{BC} = \beta _{trt} = \log (0.25)$
. The treatment effects correspond a decrease in baseline odds by 75% but with strong effect-modification, the sign of the treatment can still be flipped.
The estimand of interest is the A vs B marginal treatment effect in the BC population. Based on the current setup, the true A vs B effect is zero.
To assess the performance of population adjustment methods, we set the IPD sizes at 100, 200, or 600. We devised high, moderate, and low population overlaps by adjusting covariate distributions, each corresponding to different degrees of reductions in ESS. Factorial combinations of these factors in multivariate normal and non-normal covariate structures yield 18 scenarios. Detailed parameter configurations are in the appendix.
4.3 Methods
The following methods will be compared:
-
1. Bucher’s method
-
2. MAIC
-
3. Bayesian parametric G-computation from Remiro et al.Reference Remiro-Azócar, Heath and Baio 14
-
4. Bayesian regression marginalisation with MAIC weights
-
5. Bayesian parametric G-computation with misspecified covariate structure:
-
• With the true Normal covariate structure, the ALD population is generated from a multivariate Gamma distribution
-
• With the true non-normal covariate structure, a multivariate normal distribution is used to generate the ALD population.
-
STC and ML-NMRReference Phillippo, Dias and Ades 15 are excluded in the simulation study. This is because the STC targets the conditional estimand while we are targeting the population-adjusted marginal effect. Conversely, the operation of ML-NMR, is similar to STC in two-trial targeted comparison scenario.Reference Phillippo 22 It can target the marginal estimand by parametrising the ALD population and marginalising over it, which is equivalent to the parametric G-computation.
4.4 Performance measures
We simulate
$N_{sim}$
datasets for each scenario and estimate the treatment effect (
$\widehat {d_i}$
) of A vs B in the BC population. The true value of this effect (d) is zero. The bias of the population-adjusted estimator is the expected difference between the estimated value and the truth. Variability is measured using empirical standard errors, (the standard deviation of
$\widehat {d_i}$
across all repetitions), model average standard errors (the average of the standard error associated providing by the estimating procedure
$\widehat {\sigma }_{i,mod}$
). If these two measures are close, this reflects that the variance estimator is stable. Finally, coverage is the proportion of 95% confidence intervals that contain the true difference using Wald-type confidence intervals with
$\widehat {d}_{upper(lower),i} = 1.96\times \widehat {\sigma }_{i,mod} \pm \widehat {d}_i$
. These measures can be calculated as:
-
• Bias
$= \frac {1}{N_{sim}} \sum _{i = 1}^{N_{sim}} (\widehat {d}_i - d),$
-
• Empirical standard error
$= \sqrt {\frac {1}{N_{sim}-1} \sum _{i=1}^{N_{sim}}(\widehat {d}_i-d)^2},$
-
• Model average standard error
$= \sqrt {\frac {1}{N_{sim}} \sum _{i=1}^{N_{sim}} \widehat {\mbox {var}}(\widehat {d}_i)},$
-
• Coverage
To determine
$N_{sim}$
, the Monte Carlo standard errors (MCSE) of the performance measures should be low relative to the estimates. Since the bias and precision are of primary interests, we base the number of repetitions on the MCSE of them:
$$\begin{align*}\begin{aligned} \mbox{MCSE}_{bias} &= \sqrt{\frac{\widehat{\mbox{var}}(\widehat{d}_i)}{N_{sim}}} \\\mbox{MCSE}_{EmpSE} &= \sqrt{\frac{\widehat{\mbox{var}}(\widehat{d}_i)}{2(N_{sim}-1)}}. \end{aligned}\end{align*}$$
We consider 2,000 Monte Carlo replications for our analysis. And we conduct a further simulation with 5,000 Monte Carlo replications to ensure stability, we present results under 2,000 simulation.
5 Results
All methods produced population-adjusted effect estimates in all but one simulated scenario. In the situation where
$N_{AC} = 100$
and
$\mu _{AC} = 0.25$
, MAIC failed to find suitable weights in 43 and 44 out 2,000 replicated datasets in the Normal and non-Normal cases, respectively. However, since G-MAIC does not exhibit any estimation problems, it seems that feasible weighting solutions are available using the full IPD, but are harder to find in bootstrapped samples. This could indicate that the proposed method is more stable than the classic MAIC.
There are also 10–20 estimations that yield treatment effect estimates of negative infinity for both of the parametric G-computation methods, regardless of the covariate structures (more in the non-Normal scenario due to the slightly more skewed distribution). This is because the predicted outcome probabilities in the
$BC$
population in those datasets were exactly 1, which equals infinity after a logit transformation. This is unlikely to be an issue in practice, as these data structures are unlikely to be replicated in reality. All those problematic results were considered missing when analysing the results.
We found our findings robust to a large number of Monte Carlo replications. Here we presented the results with 2,000 Monte Carlo replications.
5.1 Bias
The bias across scenarios under the different covariate structures is shown in Figures 2 and 3. For normally distributed covariates, where a logistic regression model is implied for the trial assignment, MAIC estimates the population-adjusted estimate with minimal bias, even with small samples and poor overlap. However, the large MCSE indicates that the estimates are highly variable across the simulations. Our proposed G-MAIC recovers unbiased estimates across all scenarios and produces less variable estimates with smaller bias compared to MAIC in extreme scenarios. Bayesian parametric G-computation also has minimal bias under correctly parametrised covariate distributions but biased estimates under mis-specification, increasing as IPD sample size decreases.
Figure 2 Estimation bias under multivariate Normal covariate structure with varying overlap and sample size: from left to right average sample size reductions are 82.7% 55% and 31%.
Note: From top to bottom, methods are displayed in the order of: MAIC, G-MAIC, Bayesian Parametric G-computation, Bayesian Parametric G-computation under misspecified covariate model, Bucher’s method.
Figure 3 Esitmation bias under non-Normal covariate structure with varying overlap and sample size: from left to right average sample size reductions are 32.7% 55% and 81%.
Note: From top to bottom, methods are displayed in the order of: MAIC, G-MAIC, Bayesian Parametric G-computation, Bayesian Parametric G-computation under mis-specified covariate model, Bucher’s method.
Results change when covariates are generated under the non-normal structure. Parametric G-computation with a misspecified population remains accurate, suggesting that the multivariate normal distribution approximates the true population adequately. Nevertheless, binary covariates and non-linear dependence structure lead to the incorrect specification of implied assignment function for MAIC. As shown in 3, estimates from MAIC become biased. In the worst performing scenario—where smallest IPD sample size coincides with poor population overlap—the bias from MAIC exceeds that of unadjusted comparisons. In this scenario, the low absolute ESS, a direct result of poor overlap combined with already limited sample size, becomes further compounded by bootstrap resampling, ultimately inducing finite-sample bias within logistic regression estimates. Indeed, as illustrated by Figure 3, at similarly poor levels of overlap but with larger IPD sample sizes, the resulting bias is notably smaller. Meanwhile, the issues caused by extreme weights can be partly alleviated by a correct outcome model—G-MAIC gives estimates with much smaller bias in small samples and poorly overlapped cases, despite its slightly worse performance compared to the fully parametric methods.
5.2 Variance and variance estimation
Tables 2 and 3 show that MAIC estimates’ variability is determined by the ESS. The empirical standard error of the estimator peaks under the smallest effective sample size, regardless of covariate structure. However, G-MAIC produces more stable estimates in these extreme conditions. The variability of G-MAIC is close to the fully parametric methods. This improvement becomes more prominent as unadjusted estimates become more variable. This is due to augmentation with a regression model and the use of the full samples instead of non-parameteric bootstrapped samples for variance estimation. Therefore, re-weighted populations in G-MAIC will always have larger ESS. Additionally, using weights for averaging potential outcomes can be more stable than weighted regressions. Finally, our results indicate that dependence structures among covariates do not significantly impact variability.
Table 2 Empirical and Model SE across scenarios under multivariate Normal DGP
Table 3 Empirical and Model SE across scenarios under non-Normal DGP
We evaluate variance estimation by comparing empirical standard errors and model average standard errors. Firstly, we confirm that the variance estimation from MAIC can be unstable under poor population overlap. Comparatively speaking, variance estimation for G-MAIC is much more stable but tends to underestimate true variability— especially under small ESS—due to concentration of weights leading to an overly homogeneous ALD approximation. Fully parametric methods perform well as they extrapolate and consider the variance information when simulating the ALD.
5.3 Coverage
Figures 4 and 5 display the coverage of the 95% confidence intervals for normal and non-normal covariate structures, respectively. MAIC exhibits over-coverage in extreme scenarios due to failed variance estimation. In less extreme cases, the Normal covariate structure exhibits largely correct coverage due to minimal bias and largely correct variance estimation. Conversely, under the non-normal covariate structure, the large estimation bias can sometimes ‘over-compensate’ exaggerated model variance leading to correct coverage.
Figure 4 Coverage of 95% confidence intervals under multivariate Normal covariate structure with varying overlap and sample size: from left to right average sample size reductions are 82.7% 55% and 31%.
Note: From top to bottom, methods are displayed in the order of: MAIC, G-MAIC, Bayesian Parametric G-computation, Bayesian Parametric G-computation under misspecified covariate model, Bucher’s method.
Figure 5 Coverage of 95% confidence intervals under non-Normal covariate structure with varying overlap and sample size: from left to right average sample size reductions are 32.7% 55% and 81%.
Note: From top to bottom, methods are displayed in the order of: MAIC, G-MAIC, Bayesian Parametric G-computation, Bayesian Parametric G-computation under mis-specified covariate model,Bucher’s method.
G-MAIC shows some under-coverage mainly driven by an underestimation of true variance; thus, nominal 95% confidence interval coverage is worst with small ESS values. For fully parametric methods, the parameterically specified populations offers sufficiently good approximations to the underlying ALD covariate structure, so the coverage is close to the nominal 95%.
6 Discussion
In the real-world, regulators face increasing amounts of efficacy evidence that are based on small trials that do not compare all relevant interventions. This is especially true for interventions for rare and more severe diseases. In these scenarios, it is valuable to conduct population adjustment to enable ‘fairer’ comparisons between the available interventions. However, our simulation study demonstrates that even when the trial assignment model is correctly specified, the large variability along with unstable variance estimation, could render the standard MAIC method almost unusable. At the same time, the regulator might be reluctant to accept results from parametric G-computation since it is based on simulated covariates. Thus, we introduced G-MAIC that combines outcome regression with the weighting adjustment used in MAIC. We have shown in a comprehensive simulation study that it achieves better performance in terms of both unbiasedness and stable variance estimation that MAIC and that it is more robust under mis-specification of the trial assignment model.
The proposed G-MAIC method can offer an acceptable compromise between MAIC and parametric G-computation. With the correct trial assignment weights, G-MAIC provides an adjusted estimate with minimal bias and stable variance estimation. Apart from the efficiency gains due to the usage of a outcome model, the improved performance can also be explained by the use of Bayesian bootstrap, which provides a more stable variance estimate. G-MAIC does underestimate variance in small samples and warrants further investigations. ESS after estimating the weights can diagnose issues just as in MAIC. Analysts adopting G-MAIC could consider inflating the variance by slightly tempering the estimated weights when faced with extremely small ESS (raise the weights to a power that is smaller than 1). However, future work could focus on developing a more principled way to incorporate the uncertainty in weights estimation.
Furthermore, a bigger challenge for MAIC is the mis-specification of the trial assignment model. Under such mis-specification, the re-weighted population is no longer exchangeable with the ALD population, and mean-balancing alone is insufficient to correct all the bias from effect modifications in non-linear models. In HTA, effect-modifiers typically come in different data types, which implies that the trial assignment models can often be mis-specified. The simulation results under the non-normal data structure indicate issues with the use of MAIC in practice. We observe that low absolute ESS can exacerbate the problem non-exchangeability. Under the most extreme scenario, where the absolute ESS is drastically reduced due to the compounded effect of poor overlap and small IPD sample sizes, MAIC can produce estimates that are more biased than the unadjusted method. Previous simulation studies have assumed a multivariate Normal covariate structure, this is one of the few studies that explores how non-Normal covariate structure affects the MAIC estimates.
G-MAIC on the other hand, substantially improves over MAIC in our simulations despite using the same weights for re-weighting. This is likely due to a correctly specified outcome model. The estimate under mis-specified weights is akin to the ‘true’ treatment effect in a population that is not exchangeable with the ALD. And therefore the resulting bias depends upon the degree of non-exchangeability. It is not unimaginable that the improvements can disappear in situations where the trial populations have completely different covariate structure with different skewness. The non-exchangeability is not severe in the scenarios we tested on its own but could worsen as ESS decrease. As seen in Figure 3, the estimation bias is smaller in less extreme scenarios. In scenarios where the IPD sizes are 200 and 600, G-MAIC still offers performance gains, but to smaller extent as the ESS becomes sufficiently large.
Nevertheless, we hesitate to conclude that G-MAIC offers a perfect solution to the problem of mis-specified assignment model, as the estimand in these cases is not so interpretable. Strictly from an estimand perspective, marginalising the correct outcome relationship in a non-exchangeable population might be regarded as irrelevant to the indirect comparison problem at hand. However, we argue that all population adjustment methods suffer from this problem—all estimands are irrelevant when the covariate model is mis-specified. On a more practical level, when faced with covariate structures with potentially different skewness, one should consider adopting alternative weighting schemes that correspond to a more appropriate trial assignment model, or even matching on covariate medians in addition to the original formulation of MAIC. The connections between calibration weights and covariate structure are briefly mentioned in,Reference Wang 19 , Reference Jackson, Rhodes and Ouwens 32 and is an area for future work.
In addition, methods such as MAIC, parametric G-computation, and the proposed G-MAIC inherently produce treatment effect estimates anchored exclusively to the specific ALD trial population, substantially limiting their synthesis into larger evidence networks. As evidence accumulates and the assumption of shared effect modifiers becomes increasingly justifiable, ML-NMR emerges as a desirable alternative, as it integrates evidence from multiple IPD and ALD studies and enables the estimation of marginal or conditional effects relevant to any selected decision-making target population of interest. Nonetheless, the increased complexity, data requirements, and the stronger assumptions around shared effect-modifiers may present practical challenges. Thus, methods only applicable to pairwise indirect comparisons remain relevant in settings with early-stage evidence.
Recent work by Park et al.Reference Park, Campbell and Towle 40 has highlighted the growing interest in doubly-robust methods for population-adjusted indirect comparisons, which ensures the consistency of the estimator when either the outcome or assignment model is correctly specified. Their approach combines MAIC weights with an outcome regression model for unanchored comparisons involving survival outcomes. While the research in doubly-robust methods continue to evolve, the G-MAIC offers a flexible framework to accommodate all possible methods: the central goal of G-MAIC is to use Bayesian bootstrap to construct the ALD population non-parametrically, while any suitable outcome modelling approach can be applied in the first-stage regression.
In conclusion, we emphasise two critical aspects of population adjustment: target population and extrapolation methods. G-MAIC adopts MAIC weights, rendering estimates a ‘targeted comparison’ within the ALD trial population. This ‘fixation’ on the target population can yield different estimates for the same trial pairs, contingent on ALD/IPD availability. However, this is not a major limitation, as marginal estimands are sample-specific under effect-modifications. For G-MAIC, the target population is an interpolation between weighted and unweighted IPD. Although differ from the decision population, it might be the best compromise without additional parametric assumptions.
Extrapolation considerations are complex. G-MAIC does not extend beyond the IPD sample space, making it sensitive to population overlap but enabling the safe adoption of novel Bayesian non-parametric methods, such as Bayesian additive regression trees for outcome modelling. Extrapolations from these methods can be unreliable, relying on limited observations at the IPD covariate boundaries. In summary, future research should focus on innovative approaches to reconstruct the target population based on ALD and develop methods for robust non-linear extrapolations.
Author contributions
Conceptualization: C.G.; Methodology: C.G., A.H., G.B.; Formal analysis: C.G.; Writing—Original Draft: C.G.; Writing—Review and Editing: C.G., A.H., G.B.
Competing interest statement
The authors declare that no competing interests exist.
Data availability statement
Data availability is not applicable to this article as no new data were created or analysed in this study.
Funding statement
A.H. is funded by a Canada Research Chair in Statistical Trial Design and funded by the Discovery Grant Program of the Natural Sciences and Engineering Research Council of Canada (RGPIN-2021-03366).
Appendix
-
• The number of participants in the IPD trial,
$N_{AC} \in \{100, 200, 600\}$
. We will use a
$1:1$
allocation ratio for both sample sizes; -
• The covariate structures of IPD in both trials:
-
– The Normal case: five continuous covariates with each covariate k in trial j generated from a Normal distribution:
$\boldsymbol{X_{jk}} \sim \mbox {Normal}(\mu _{jk}, \sigma _{jk}^2)$
. The pairwise correlation
$\rho $
has that
$\rho = 0.25$
-
– The non-Normal case: starting with a Normally distributed covariate
$X_1 \sim \mbox {Normal}(\mu _1, \sigma _1^2)$
, a binary covariate
$X_2 \sim \mbox {Bernoulli}(p_2)$
is generated such that
$\mbox {logit}(p_2) = -0.5 + 0.5X_1$
. The third covariate
$X_3$
depend on
$X_2$
with
$X_3 \sim \mbox {Normal}(-0.5X_2, 1)$
.
$X_4$
is generated from a Gamma distribution with mean
$\mu _4 = |0.2X_1 + 0.3X_3|$
, and dispersion parameter
$d_4 = 0.5$
. Finally, a binary
$X_5 \sim \mbox {Bernoulli}(p_5)$
is generated with
$\mbox {logit}(p_5) = 0.1X1 - 0.1X2 + 0.05X3$
.
-
-
• The overlap between trials: overlap between multivariate distributions is complex and hard to compute. Instead, we choose the parameter values for the distributions to directly target a corresponding reduction in ESS. This is achieved by repeatedly applying MAIC to 2,000 simulated
$AC$
and
$BC$
populations with
$N_{AC} = N_{BC} = 600$
, and compute the average reduction ESS across all repetitions. In the Normal case, we have
$\boldsymbol{\mu }_{BC} = 0.6$
. The population standard deviation is fixed to be
$sd_X = 0.4$
for both trials. The marginal mean in the
$AC$
population is set at
$\mu _{AC} \in \{0.45,0.375,0.25\}$
. corresponding to 31%, 54% and 82.7% reduction in ESS. In the second scenario, we set
$\mu _1^{AC} = 0, \sigma _{1,AC}^2 = 0.2, \sigma _{1,BC}^2 = 0.4$
and
$\mu _1^{BC} \in \{0.275,0.4,0.6\}$
, corresponding to
$32.7\%, 55\%$
and
$82.7\%$
reduction in ESS.
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