Table 1 Model characteristics for one two-stage and five one-stage models for estimating treatment covariate interaction in an individual participant meta-analysis
Model | Equation | Modelling assumptions |
|---|---|---|
Two stage model: In the first stage, maximum likelihood regression model is used within each trial (Simmonds and Higgins 2007 [4]), including a treatment effect and a treatment-covariate interaction term. In the second stage, the interaction effect estimates from each trial (\({\hat{\upgamma}}_i\)) are combined using conventional meta-analysis techniques (in this case, the inverse-variance meta-analysis using the DerSimonian-Laird random effect method), producing a summary treatment-covariate interaction estimate. | ||
Meta-analysis of interactions (Simmonds and Higgins 2007 [4]) | g(yij) = Φi + θixij + μizij + γixijzij | ● The studies are estimating a different, yet related interaction effects. |
One-stage models: A one-stage maximum likelihood regression model includes both a treatment effect and a treatment-covariate interaction term, with data from all studies in the same model. The common effect version of the model is as equation for meta-analysis of interactions, except now the parameters are assumed common across all studies. A separate intercept term (Φ𝑖) retains distinctions between studies, avoiding the assumption that data arise from one ‘mega trial’ | ||
Common interaction effect: model (Tuner et al. 2000 [14]) | \({\displaystyle \begin{array}{c}\textrm{g}\left({y}_{ij}\right)={\Phi}_i+\left(\theta +{u}_i\right)\ {x}_{ij}+\mu {z}_{ij}+\gamma {x}_{ij}{z}_{ij}\\ {}{u}_i\sim N\left(0,{\tau}^2\right)\end{array}}\) | ● The true effect of the treatment is allowed to vary between studies. ● The true effect of the interaction is assumed common between studies. |
Common interaction effect: model 2 (Jackson et al. 2018 [6]) | \({\displaystyle \begin{array}{c}\textrm{g}\left({y}_{ij}\right)=\left(\Phi +{v}_i\right)+\left(\theta +{u}_i\right)\ {x}_{ij}+\mu {z}_{ij}+\gamma {x}_{ij}{z}_{ij}\\ {}\left(\begin{array}{c}{u}_i\\ {}{v}_i\end{array}\right)\sim N\left(\left(\begin{array}{c}0\\ {}0\end{array}\right),\left(\begin{array}{cc}{\tau}_{\theta}^2& \lambda \\ {}\lambda & {\tau}_{\phi}^2\end{array}\right)\right)\end{array}}\) | ● The true effect of the treatment is allowed to vary between studies. ● The true effect of the interaction is common between studies. ● The random effects for the trial and treatment are correlated. |
Common interaction effect: model 3 (Jackson et al. 2018 [6]) | \({\displaystyle \begin{array}{c}\textrm{g}\left({y}_{ij}\right)=\left(\Phi +{v}_i\right)+\left(\theta +{u}_i\right)\ {x}_{ij}+\mu {z}_{ij}+\gamma {x}_{ij}{z}_{ij}\\ {}\left(\begin{array}{c}{u}_i\\ {}{v}_i\end{array}\right)\sim N\left(\left(\begin{array}{c}0\\ {}0\end{array}\right),\left(\begin{array}{cc}{\tau}_{\theta}^2& \lambda \ast \\ {}\lambda \ast & {\tau}_{\phi}^2\end{array}\right)\right)\end{array}}\) *λ = 0 | ● The true effect of the treatment is allowed to vary between studies. ● The true effect of the interaction is common between studies. ● The random effects for the trial and treatment are uncorrelated. |
Random interaction: | \({\displaystyle \begin{array}{c}\textrm{g}\left({y}_{ij}\right)=\left(\Phi +{v}_i\right)+\left(\theta +{u}_i\right)\ {x}_{ij}+\mu {z}_{ij}+\left(\gamma +{w}_i\right){x}_{ij}{z}_{ij}\\ {}\left(\begin{array}{c}{u}_i\\ {}{v}_i\\ {}{w}_i\end{array}\right)\sim N\left(\left(\begin{array}{c}0\\ {}0\\ {}0\end{array}\right),\left(\begin{array}{ccc}{\tau}_{\theta}^2& 0& 0\\ {}0& {\tau}_{\phi}^2& 0\\ {}0& 0& {\tau}_{\gamma}^2\end{array}\right)\right)\end{array}}\) | ● The true effect of the treatment is allowed to vary between studies. ● The true effect of the interaction is allowed to vary between studies. ● The random effects for the trial, treatment and interaction are uncorrelated. |
Within study model | \({\displaystyle \begin{array}{c}\textrm{g}\left({y}_{ij}\right)=\left(\Phi +{v}_i\right)+\left(\theta +{u}_i\right)\ {x}_{ij}+\mu {z}_{ij}+\xi {x}_{ij}\left({z}_{ij}-{\overline{z}}_i\right)+\upeta {\overline{z}}_i\\ {}\left(\begin{array}{c}{u}_i\\ {}{v}_i\end{array}\right)\sim N\left(\left(\begin{array}{c}0\\ {}0\end{array}\right),\left(\begin{array}{cc}{\tau}_{\theta}^2& 0\\ {}0& {\tau}_{\phi}^2\end{array}\right)\right)\end{array}}\) \({\overline{z}}_i\) is the average covariate value in trial i, so ξ is the parameter for the within-trial interaction. | ● The effect of the treatment and covariates are assumed common between studies. ● Only the within-study information on the treatment-covariate interaction is used, avoiding the assumption that the observed across-study relationships do reflect the individual-level relationships within trials. |