Introduction
Methods
Study design and data structure
Model 1: three level Bayesian hierarchical model for percent weight loss
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α0ij = α0i0 + γj, where \({\gamma}_j\sim N\left(0,{\sigma}_{\gamma}^2\right)\) is patient level variation.
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α0i0 = α000 + ηi, where ηi ~ \(N\left(0,{\sigma}_{\eta}^2\right)\) is site level variation and a000 is the model intercept.
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ϵijt~N(0, σ2) is within patient residual error.
Model 2: Bayesian hierarchical model for percent weight loss with group assignment as a partially nested effect
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α0ikj = α0ik0 + γj, where \({\gamma}_j\sim N\left(0,{\sigma}_{\gamma}^2\right)\) represents the patient level variation.
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α0ik0 = α0i00 + ϑk, where \({\vartheta}_k\sim N\left(0,{\sigma}_{\vartheta}^2\right)\) represents the intervention group level variation for participants in the two group-based arms and for participants in the in-clinic individual arm ϑ0 = 0.
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α0i00 = α0000 + ηi, ηi ~ \(N\left(0,{\sigma}_{\eta}^2\right)\) represents the site level variation and a0000 is the intercept.
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ϵikjt~N(0, σ2) is the within patient residual error
Quantities of interest
Computation and software
Results
Model convergence assessment and predictive checking
Model result
Model 1
Mean | Standard Deviation | 2.50% | 97.50% | ||
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Intercept | a000 | 6.24 | 0.54 | 5.18 | 7.29 |
In-clinic group effect (ref: in-clinic indiv) | β1 | 2.67 | 0.69 | 1.36 | 4.04 |
Phone-based group effect (ref: in-clinic indiv) | β2 | 1.98 | 0.69 | 0.64 | 3.34 |
18 months effect (ref: 6 months) | β3 | −2.26 | 0.28 | −2.8 | −1.72 |
24 months effect (ref: 6 months) | β4 | −3.05 | 0.28 | −3.6 | − 2.52 |
In-clinic group effect * 18 months effect | β5 | 0.06 | 0.61 | −1.15 | 1.26 |
Phone-based group effect *18 months effect | β6 | −2.16 | 0.79 | −3.7 | − 0.62 |
In-clinic group effect * 24 months effect | β7 | −0.41 | 0.38 | −1.16 | 0.34 |
Phone-based group effect * 24 months effect | β8 | −0.1 | 0.4 | −0.87 | 0.68 |
Affiliation: Marshfield Clinic (ref: KUMC) | β9 | −0.83 | 0.39 | −1.58 | −0.07 |
Affiliation: UNMC (ref: KUMC) | β10 | −0.51 | 0.39 | −1.28 | 0.28 |
Observation level variation | σ | 3.92 | 0.06 | 3.8 | 4.03 |
Site level variation | ση | 0.98 | 0.36 | 0.22 | 1.67 |
Patient level variation | σr | 6.66 | 0.15 | 6.38 | 6.95 |
Model 2
Mean | Standard deviation | 2.50% | 97.50% | ||
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Intercept | a000 | 6.25 | 0.53 | 5.2 | 7.28 |
In-clinic group effect (ref: in-clinic indiv) | β1 | 2.67 | 0.7 | 1.31 | 4.05 |
Phone-based group effect (ref: in-clinic indiv) | β2 | 1.94 | 0.7 | 0.53 | 3.34 |
18 months effect (ref: 15 months) | β3 | −2.26 | 0.28 | −2.8 | − 1.71 |
24 months effect (ref: 15 months) | β4 | −3.06 | 0.28 | −3.61 | − 2.51 |
In-clinic group effect * 18 months effect | β5 | 0.09 | 0.62 | −1.12 | 1.32 |
Phone-based group effect * 18 months effect | β6 | −2.13 | 0.81 | −3.73 | −0.55 |
In-clinic group effect * 24 months effect | β7 | −0.41 | 0.39 | −1.19 | 0.35 |
Phone-based group effect * 24 months effect | β8 | −0.1 | 0.39 | −0.88 | 0.66 |
Affiliation: Marshfield Clinic (ref: KUMC) | β9 | −0.83 | 0.39 | −1.59 | −0.06 |
Affiliation: UNMC (ref: KUMC) | β10 | −0.51 | 0.39 | −1.27 | 0.26 |
Observation level variation | σ | 3.92 | 0.06 | 3.81 | 4.03 |
Site level variation | ση | 0.94 | 0.37 | 0.15 | 1.66 |
Group level variation | σϑ | 1.28 | 0.44 | 0.28 | 2.08 |
Patient level variation | σr | 6.59 | 0.15 | 6.29 | 6.89 |
Looic (se) | WAIC (se) | |
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Model 1 | 22,025.1 (127.4) | 21,816.1 (122.6) |
Model 2 | 22,016.7 (126.2) | 21,813.7 (122.2) |
Model 1 – Model 2 | 8.4 (4.6) | 2.4 (3.0) |