Table 1 Individual patient data indirect comparison methods

• Adjusted indirect comparison: The method derives an indirect estimate for the relative effectiveness or safety of two different treatments adjusted by comparing the results of their direct comparisons (i.e. pairwise meta-analyses) with a common comparator treatment [68]. Consider, for example, a tree-shaped triangular network composed by some IPD (or IPD and aggregated data) studies comparing treatment A against treatment B, and some studies comparing treatment A against treatment C. The method uses the summary treatment effect estimates derived by a pairwise meta-analysis (which can be a one-stage or two-stage approach) for studies Avs.B and for studies Avs.C.

• Matching adjusted indirect comparison (MAIC): The method estimates an indirect comparison of the treatments of interest [48]. Consider the tree-shaped triangular network ABC composed by some IPD studies comparing Avs.B treatments and some aggregated data studies comparing Avs.C treatments. The method uses the information from the IPD trials on one treatment arm (B) and the information from the aggregated data trials on the other treatment of interest (C). The patient characteristics from the IPD trials on treatment B are then matched to the ones of the aggregated data trials on treatment C using an approach similar to propensity score weighting. Specifically, the patient baseline characteristics in IPD trials and treatment B are reweighted so that the weighted average of the patient characteristics matches the characteristics of the population in treatment C of the aggregated data trials. The weights are modeled as a linear combination of all reported baseline characteristics. After matching the baseline characteristics between the two groups, the treatment outcomes are compared across the trial populations using the adjusted mean for treatment B and observed mean for treatment C.

• Simulated treatment comparison (STC): A similar approach to the MAIC is the STC, which uses a different process to adjust for population characteristics [49]. Considering the same tree-shaped ABC network, the STC method estimates the treatment response in B using information from the IPD trials and a predictive regression model with patient-level characteristic covariates. Then a statistical calibration of the trial(s) with IPD is performed to match the characteristics with aggregated data trials and treatment C. Trial data are simulated for treatment B based on the statistical calibration. The adjusted mean for treatment B is compared with the observed mean for treatment C. The MAIC and STC methods may be particularly useful when there is insufficient data from head-to-head comparison trials, and when there is insufficient data to apply an adjusted indirect comparison (e.g., disconnected network of trials) [50].

• Network meta-analysis (NMA) approaches: When both direct and indirect evidence (IPD or in combination with aggregated data or aggregated data only) are available for the same comparison (e.g., Bvs.C), then these may be combined in a mixed effect size using the mixed comparison method [69]. The mixed comparison estimate is a weighted average of the meta-analytic effect estimate Bvs.C and the adjusted indirect comparison for Bvs.C. A suggested approach for combining direct and indirect evidence using the mixed comparison method is the inverse variance method with weights the inverse of the variance of the estimated effects. An approach to simultaneously compare multiple treatments in a single analysis is by using a meta-regression model [70]. Each study-specific treatment effect is expressed as a linear function of the basic parameters, which is a set of comparisons of the treatments in the network versus the reference treatment. Assuming A is the reference treatment then Avs.B and Avs.C are the basic parameters for the ABC network. This approach uses the different treatment comparisons as covariates in the meta-regression model. In particular, it uses dummy variables for the basic parameters to define the basic contrasts Avs.B and Avs.C, omits the intercept, and specifies the covariate values so that consistency between direct and indirect evidence holds. An alternative way to apply a NMA is by using hierarchical models [71]. Studies have shown that the majority of NMA applications have been carried out in a Bayesian setting using hierarchical models [4, 10, 42].