Fig 1.
Visualization of potential outcomes and treatment effect for a patient in an antidepressant trial.
The patient is randomized to either the placebo or the active arm, corresponding to two hypothetical “potential outcomes” (Red scores). Only one outcome can be observed, as a patient cannot receive both interventions simultaneously. The difference between the two outcomes is the “individual treatment effect” of the intervention (Dark green score). The individual treatment effect is unobservable and can be imaged to be drawn from hypothetical distributions of the treatment effect. The variance of this distribution corresponds to the treatment effect heterogeneity. The factor ρ is the correlation between the placebo response and the individual treatment effect. All numbers signify depression severity on the HAMD-17 scale.
Fig 2.
PRISMA 2009 flow diagram.
Fig 3.
Causal graph depicting relationship between treatment, mean and SD.
The treatment Tx has a causal effect of around 2 Hamilton points on the mean of the study population. If we control for the mean ~ SD relationship when estimating the variability between treatment and control, we only measure the direct effect of treatment on variability, while no such control (as with the VR effect size) measures the total effect of the treatment on outcome variability.
Fig 4.
Linear association between lnMean and lnSD.
Linear association between lnMean and lnSD using a varying intercept model, where the intercepts were allowed to vary between studies with different depression scales. Red dots represent active groups, blue dots represent placebo groups.
Fig 5.
Posterior credible intervals for total effect, direct effect and indirect effect as determined by the REMR.
eα represents the study-specific direct effect, RRβ the study-specific indirect effect, eμ is the meta-analytic mean of the direct effect.
Fig 6.
Posterior credible intervals for the eμ parameter for the different models.
REMA: random-effects meta-analysis. FEMA: fixed-effects meta-analysis. REMR: random-effects meta-regression. The lnCVR underestimates the variability in the active group, while the results are very similar for the REMR and the lnVR meta-analysis.
Fig 7.
Widely applicable information criterion (WAIC) depicted on a logarithmic scale.
Higher values signify a better predictive fit of the underlying model. Bars indicate standard errors. REMA: random-effects meta-analysis. FEMA: fixed-effects meta-analysis. REMR: random-effects meta-regression.
Fig 8.
Influence of baseline severity on outcome variability.
Estimated VR of antidepressants versus placebo as a function of baseline severity on the HAMD17-scale as determined by the REMR. Depicted are the mean estimates and 95% HDIs for three different values of the response ratio (mean response active versus mean response placebo). The meta-analytic mean of RR is around 1.22 in the Cipriani data set.
Fig 9.
VR for different antidepressant classes.
Posterior credible intervals for the eμ parameter for different antidepressant classes as determined by the random-effect meta-regression.
Table 1.
Upper bound on treatment effect heterogeneity compatible with the results.
Fig 10.
Change score of 1000 simulated patients.
Boxplot (a) of change score under placebo (blue) and under active treatment (red) for ρ = - 0.4, SDte = 6.35 HAMD-17 points and VR = 1.02. (b) depicts pre-post change for each individual patient, gauged to 0. Note, that in this particular simulation, the SDte is not exactly equal to 6.5, as all simulations contain random processes.
Fig 11.
Potential outcomes and individual treatment effect of 100 simulated patients.
Potential outcome under placebo and under active treatment of 100 simulated patients, ρ = - 0.4, SDTE = 6.35 and VR = 1.2. Slopes represent individual treatment effect, which varies substantially in this simulation. Blue lines indicated improvement under active treatment, red lines deterioration.