Table 1 Summary of simulation scenarios for evaluating the performance of BQQ plots
Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | |
|---|---|---|---|---|
Model: | 1 (Equation 4) | 1 (Equation 4) | 2 (Equation 5) | 2 (Equation 5) |
Number of patients (N) | {30, 50, 100, 150, 200, 300, 500} | {20, 60, 100, 160, 200, 300, 500} | {30, 50, 100, 150, 200, 300, 500} | {30, 50, 100, 150, 200, 300, 500} |
# of measurements per patient, n = k0,ω + k1,ω | For all scenarios, when k1,ω = 2, n = 4; when k1,ω = 4, n = 6. | |||
Binary covariate | xω~Bernoulli(0.6) (for all scenarios) | |||
Measurement errors | \( \mathrm{i}.\mathrm{i}.\mathrm{d}.,{\varepsilon}_{\omega, j}\sim N\left(0,{\sigma}_{\varepsilon}^2=10\right) \) (for all scenarios) | |||
Fixed effects (ψ) | (21, 2, −5, 0.5)T | (21, 2, −5, 0.5)T | (21.4, 1.92, −3.97, 0.35)T | (24, 1.92, 0.97, −0.35)T |
Non-normal random effects | \( {\boldsymbol{\tau}}_{\omega}\sim \frac{1}{2}N\left({\boldsymbol{m}}_1^{\ast }=w\bullet {\boldsymbol{m}}_1,V\right) \) \( +\frac{1}{2}N\left({\boldsymbol{m}}_2^{\ast }=w\bullet {\boldsymbol{m}}_2,V\right) \) m1 = (0, −1)T, m2 = (0, 1)T \( V=\left[\begin{array}{cc}1& 0.9\\ {}0.9& 1\end{array}\right] \) w ∈ {1, 2, 3, 4, 5} | \( {\boldsymbol{\tau}}_{\omega}\sim \frac{3}{4}N\left({\boldsymbol{m}}_1,V\right)+\frac{1}{4}N\left({\boldsymbol{m}}_2,V\right) \) m1 = (0, −1)T, m2 = (0, 3)T \( V=\left[\begin{array}{cc}{\sigma}_1^2& 0.9\\ {}0.9& {\sigma}_2^2\end{array}\right] \) \( {\sigma}_1^2={\sigma}_2^2\in \left\{1,2,3,4,5\right\} \) | τω~tv(m, Γ) v ∈ {3, 5, 7, 9, 11, 13} m = (0, 0, 0)T \( \Gamma =\left[\begin{array}{ccc}10.4& 0.279& -0.341\\ {}0.279& 13.06& -2.466\\ {}-0.341& -2.466& 0.581\end{array}\right] \) | \( {\boldsymbol{\tau}}_{\omega}\sim \frac{1}{2}N\left({\boldsymbol{m}}_1^{\ast }=w\bullet {\boldsymbol{m}}_1,V\right) \) \( +\frac{1}{2}N\left({\boldsymbol{m}}_2^{\ast }=w\bullet {\boldsymbol{m}}_2,V\right) \) m1 = (0, −1, 1)T m2 = (0, 1, −1)T \( V=\left[\begin{array}{ccc}10.4& 0.279& -0.341\\ {}0.279& 13.06& -2.466\\ {}-0.341& -2.466& 0.581\end{array}\right] \)w ∈ {0.5, 1, 1.5, 2, 2.5, 3} |
Reference normal random effectsa | τω~N(m, D∗) m = (0, 0)T \( {D}^{\ast }=\frac{1}{2}{\boldsymbol{m}}_1^{\ast }{{\boldsymbol{m}}_1^{\ast}}^T+\frac{1}{2}{\boldsymbol{m}}_2^{\ast }{{\boldsymbol{m}}_2^{\ast}}^T+V \) | τω~N(m, D∗) m = (0, 0)T \( {D}^{\ast }=\frac{3}{4}{\boldsymbol{m}}_1{\boldsymbol{m}}_1^T+\frac{1}{4}{\boldsymbol{m}}_2{\boldsymbol{m}}_2^T+V \) | τω~N(m, D∗) m = (0, 0, 0)T \( {D}^{\ast }=\left(\frac{v}{v-2}\right)\Gamma \) | τω~N(m, D∗) m = (0, 0, 0)T \( {D}^{\ast }=\frac{1}{2}{\boldsymbol{m}}_1^{\ast }{{\boldsymbol{m}}_1^{\ast}}^T+\frac{1}{2}{\boldsymbol{m}}_2^{\ast }{{\boldsymbol{m}}_2^{\ast}}^T+V \) |