Bayesian splines versus fractional polynomials in network meta-analysis

Background Network meta-analysis (NMA) provides a powerful tool for the simultaneous evaluation of multiple treatments by combining evidence from different studies, allowing for direct and indirect comparisons between treatments. In recent years, NMA is becoming increasingly popular in the medical literature and underlying statistical methodologies are evolving both in the frequentist and Bayesian framework. Traditional NMA models are often based on the comparison of two treatment arms per study. These individual studies may measure outcomes at multiple time points that are not necessarily homogeneous across studies. Methods In this article we present a Bayesian model based on B-splines for the simultaneous analysis of outcomes across time points, that allows for indirect comparison of treatments across different longitudinal studies. Results We illustrate the proposed approach in simulations as well as on real data examples available in the literature and compare it with a model based on P-splines and one based on fractional polynomials, showing that our approach is flexible and overcomes the limitations of the latter. Conclusions The proposed approach is computationally efficient and able to accommodate a large class of temporal treatment effect patterns, allowing for direct and indirect comparisons of widely varying shapes of longitudinal profiles.


Table and Figures for Additional file 1
Additional file 1: Table S6 contains simulated temporal patterns for additional simulation scenarios (i) linear, (ii) logarithmic, (iii) piecewise linear monotonic, (iv) mix of the previous with one treatment effect being constant. Figures S12, S13, S14 and S15 show respective estimated profiles obtained for the B-spline, P-spline and FP models, along with true values used to simulate the data. Figure S16 shows estimated profiles for case (iii) with binary outcomes. Figure S17 shows estimated profiles for a scenario, as in the Simulation Study Section, in which temporal patterns have been generated according to the mixed treatment comparison (MTC) model. Figure S18 shows estimated profiles for a scenario in which temporal patterns have been generated from the Bayesian evidence synthesis techniques -integrated twocomponent prediction (BEST-ITP) model. Figure S19 illustrates the influence of a non-closed network. Table S6 Temporal patterns for treatment effects of treatments A,B and C used in the simulations shown in Figures S12-S15. The mixed scenario contains a constant treatment effect for treatment B and a quadratic effect for treatment C. The simulated temporal patterns γ At , γ Bt and γ Ct in the scenarios are (i) linear, (ii) logarithmic, (iii) piecewise linear monotonic, and (iv) a mix of the above with one of the treatment effects being constant.
Figure S12 Posterior estimated profiles obtained for the linear scenario. Red lines indicate the values used to simulate the data. Estimates and 95% credible intervals obtained from the respective models are indicated in blue (FPs), black (B-splines) and green (P-splines). Due to their small size, credible intervals are not visible.

Figure S13
Posterior estimated profiles obtained for the logarithmic scenario. Red lines indicate the values used to simulate the data. Estimates and 95% credible intervals obtained from the respective models are indicated in blue (FPs), black (B-splines) and green (P-splines). Due to their small size, credible intervals are not visible for B-splines and P-splines.

Figure S14 Posterior estimated profiles obtained for the piecewise linear monotonic scenario.
Red lines indicate the values used to simulate the data. Estimates and 95% credible intervals obtained from the respective models are indicated in blue (FPs), black (B-splines) and green (P-splines). Due to their small size, credible intervals are not visible for B-splines and P-splines.

Figure S15 Posterior estimated profiles obtained for the mixed scenario.
Red lines indicate the values used to simulate the data. Estimates and 95% credible intervals obtained from the respective models are indicated in blue (FPs), black (B-splines) and green (P-splines). Due to their small size, credible intervals are not visible for B-splines and P-splines.

Figure S16 Posterior estimated profiles obtained for the piecewise linear scenario with binary outcomes.
Red lines indicate the values used to simulate the data. Estimates and 95% credible intervals obtained from the respective models are indicated in blue (FPs), black (B-splines) and green (P-splines). Due to their small size, credible intervals are not visible for B-splines and P-splines.

Figure S17 Posterior estimated profiles obtained for the MTC scenario.
Red lines indicate the values used to simulate the data. Estimates and 95% credible intervals obtained from the respective models are indicated in blue (FPs), black (B-splines) and green (P-splines). Due to their small size, credible intervals are not visible for B-splines and P-splines. Mean outcomes are modelled as θ sjt = μst + δ sj , the sum of a study-and time-specific effect across treatment arm and a study-specific arm deviation, where μst ∼ Normal (−s, 1) and δ sj ∼ Normal (0.5, 1), i.e., in this model relative treatment effects are assumed to be constant over time. Simulation data is generated with σ sjt = 1.2/ √ ns, i.e., assuming no within-study correlation between subsequent time points.

Figure S18 Posterior estimated profiles obtained for the BEST-ITP scenario.
Red lines indicate the values used to simulate the data. Estimates and 95% credible intervals obtained from the respective models are indicated in blue (FPs), black (B-splines) and green (P-splines). Due to their small size, credible intervals are not visible for B-splines and P-splines. Mean outcomes are modelled as θ sjt = (φs + δ j )(1 − e −p j t )/(1 − e −p j T ), where φs ∼ Normal (−3, 1), and δ 1 = 0, δ 2 = 0.5, δ 3 = 1 are treatment effects at the end follow-up time T , to which they increase according the shape given by the exponential function factor with p 1 = −0.1, p 2 = −0.15 and p 3 = −0.15.

Figure S19
Posterior estimated profiles obtained for the piecewise linear monotonic scenario (iii) under the non-closed network in Figure ?? (right graph). The additional treatment D, which the extra study 4 directly compares with treatment B, is given by the temporal effect pattern γ Dt = −t for t ∈ [0, 4], γ Dt = −3t/4 − 1 for t ∈ (4, 8], γ Dt = −t/4 − 5 for t ∈ (8, 12] and γ Dt = −t/12 − 7 for ∈ (12, 24]. Follow-up times of study 4 are placed at weeks 4, 8, 12 and 24, the number of observations is chosen as n 4 = 110, and the variance as τ 2 4 = 3. Red lines indicate the values used to simulate the data. Estimates and 95% credible intervals obtained from the respective models are indicated in blue (FPs), black (B-splines) and green (P-splines). Due to their small size, credible intervals are not visible for B-splines and P-splines.