We present the \(_\) for the network of 18 antidepressants for the acute treatment of adults with major depressive disorder. We show how treatment performance varies for different values of the trade-off \(\lambda\).

We first estimated for each treatment the absolute probabilities \(_\) of response or risk for the outcomes as described in Equations (1) and (2) of Additional file 3. Fluoxetine was chosen as the reference drug, so we estimated the odds in this control group by meta-analysing the Fluoxetine arms.

The probability \(_}\) for response, remission, dropouts for any cause, and due to side effects were 0.569, 0.347, 0.236, and 0.078, respectively, as reported in Table 1.

After consulting with clinicians, we gave a weight of 0.3 and 0.7 to the response and remission outcomes respectively; and weights of 0.7 and 0.3 to dropout due to side effects and dropout due to any cause outcomes, respectively.

The \(_\) values for different \(\lambda\) are shown in Additional file 4 and illustrated in Fig. 1. In line with Eq. (2), the \(_\) values decrease with increasing values of \(\lambda\). However, this decrease is less pronounced for some treatments, such as vortioxetine which, for the chosen weights, seems to retain its high performance for any trade-off between beneficial and harmful outcomes. Whatever the trade-off between benefits and risks, reboxetine remains the worst-performing drug, while vortioxetine, bupropion, and escitalopram are consistently the best options.

\(_\) for the network of 18 antidepressants for the acute treatment of major depressive disorder. The benefit spie chart included response and remission with weights 0.3 and 0.7, respectively, and the harm spie chart included dropout due to side effects and due to any cause with weights 0.7 and 0.3, respectively

To transform the efficacy outcome, overall symptoms of schizophrenia, to the same scale, we selected a representative study that measures the change in symptoms on the PANSS scale [15]. The mean endpoint \(}_.\mathrm}\) and standard deviation \(}_.\mathrm}\) for Placebo, to be used in Equations (3) and (4) of Additional file 3 to obtain the absolute mean score, were 98.4 and 21.4, respectively. Since the outcomes must be between 0 and 1, we have standardised the absolute mean score using the formula \(_= \frac_-30}\) (PANSS score can range between 30 and 210), as described in Equation (5) of Additional file 3. Then, the obtained value was reversed so that higher values equate to better outcomes (\(1-_\)).

The absolute probabilities \(_\) of risk for antiparkinson medication use were obtained using Equations (1) and (2) of Additional file 3 by estimating the odds for placebo by meta-analysing the reference arms; \(_}\) was estimated to be 0.093 as reported in Additional file 5.

The absolute risk probabilities \(_\) for the remaining harmful outcomes, weight gain, prolactin elevation, and QTc prolongation, were converted from the corresponding continuous outcomes using Equation (6) of Additional file 3. To derive the control group probabilities \(_}\) (Equation (7) of Additional file 3) we used the dichotomous variables to distinguish patients with and without the response based on a cut-off C of at least 7% for weight gain and study-specific thresholds for prolactin elevation and QTc prolongation. The estimated \(_}\) values were 0.034, 0.019, and 0.006, for weight gain, prolactin elevation, and QTc prolongation, respectively. The obtained probabilities and corresponding SMDs for each treatment are available in Additional file 5. Due to missing data for one or more outcomes, 18 antipsychotics were not included in the calculation of the \(}_\).

After consulting with clinicians, we gave a weight of 0.4 and 0.3 to antiparkinson medication use and weight gain, respectively, to reflect the importance of these safety outcomes compared to the other two which were both given a weight of 0.15. The \(}_\) values for different \(\lambda\) values are shown in Additional file 6 and illustrated in Fig. 2.

\(_\) quantity for the network of antipsychotics for the acute treatment of multi-episode schizophrenia. The benefit spie chart included only one efficacy outcome, overall symptoms of schizophrenia, and the harm spie chart included antiparkinson medication use, weight gain, prolactin elevation, and QTc prolongation with weights 0.4, 0.3, 0.15, and 0.15, respectively

The decrease of \(}_\) is particularly evident for haloperidol which, for the chosen weights, goes from being among the best treatments to being the worst active drug when the willingness to tolerate harm decreases. Whatever the trade-off between benefits and harms, placebo remains the least preferable treatment, while amisulpride, olanzapine, and risperidone remain the most preferable.

For this example, it was not possible to estimate absolute probabilities or use any of the conversion methods described previously due to the variety of scales used to calculate this score and the lack of a specific cut-off to define responders using these scales. Therefore, for all outcomes, we first estimated the relative treatment effects (SMD for continuous outcomes and OR for the safety outcome) of each intervention versus placebo and then, produced the relative SUCRA that we employed as the outcome measures \(_\) (Equation (1)) to plot in the spie charts. For the safety outcome, any adverse event, we calculated the SUCRA to reflect the fact that in the spie chart framework, higher values for the safety outcomes must indicate higher harm, as previously explained. Therefore, the corresponding SUCRA ranking is reversed compared to what the ordinary SUCRA ranking for a safety outcome would look like, i.e. the best-performing treatment in terms of rate of adverse events will have the lowest SUCRA value in our case, instead of the highest value, as it would usually be. The calculated SUCRA values are reported in Additional file 7.

We calculated \(_\) by giving a weight of 0.5 to both efficacy outcomes, i.e. social-communication difficulties and repetitive behaviours. Due to missing data for one or two outcomes, 16 interventions were not included in the calculation of the \(_\).

The \(_\) values for different trade-off values are reported in Additional file 8 and illustrated in Fig. 3. The \(}_\) decrease, for increasing values of \(\lambda\), is nearly null for some treatments, such as folinic acid, sapropterin, and sertraline. However, for some treatments the decrease in the \(}_\) values is so large that they shift from being the most efficacious treatments to being among the least beneficial ones, particularly risperidone and guanfacine.

\(_\) quantity for the network of pharmacological and dietary‑supplement treatments for autism spectrum disorder. The benefit spie chart included two efficacy outcomes, changes in core symptoms for social-communication difficulties, and repetitive behaviours with weights 0.5 each, and the safety spie chart included one outcome, any adverse event

Unlike the previous examples, the range of \(}_\) for this example is quite broad, also including negative values. This is because the quantities here are calculated from SUCRA which estimates the probability that a treatment X outperforms its competitors Y, Z, etc., and hence depends on the performance of the competitors Y, Z, etc. Consequently, SUCRA values can be large (e.g. above 0.7) for “high-performing” interventions, even when the outcome is rare (as in safety outcomes). Therefore, the SUCRA values for harm outcomes, when included in the spie charts and, in turn, in the \(_\) formula, could then produce negative values if the corresponding SUCRAs for positive outcomes are not of the same magnitude.