Table 1 Overview of the simulation study

Step 1

Choose a random effects distribution f(ψ) from candidates including normal distributions, skew-normal distributions, shifted exponential distributions and shifted log-normal distributions, where ψ represents a true parameter vector of the random effects distribution.

Step 2

Choose the number of studies (k), mean of the distribution for the within-study variance (σ ²) and true parameters of the random effects distribution (ψ).

Step 3

Draw a within-study variance of the treatment effect estimate for the ith study (i=1,…,k); \(\tilde {\sigma }_{i}^{2}\sim N(\sigma ^{2},0.040)\) conditioned on \(0.010<\tilde {\sigma }_{i}^{2}<(2\sigma ^{2}-0.010)\).

Step 4

Draw a sampling error of the treatment effect estimate for the ith study (i=1,…,k); \(\tilde {\epsilon }_{i}\sim N(0,\tilde {\sigma }_{i}^{2})\), where \(\tilde {\sigma }_{i}^{2}\) is obtained in Step 3.

Step 5

Draw a true treatment effect for the ith study (i=1,…,k); \(\tilde {\theta }_{i}\sim f(\psi)\), where f(ψ) is the specified random effects distribution with the true parameter ψ.

Step 6

Obtain a treatment effect estimate for the ith study (i=1,…,k); \(\tilde {y}_{i}=\tilde {\theta }_{i}+\tilde {\epsilon }_{i}\), where \(\tilde {\epsilon }_{i}\) and \(\tilde {\theta }_{i}\) are obtained in step 4 and step 5 respectively.

Step 7

Using \(\tilde {y}_{i}\) and \(\tilde {\sigma }_{i}^{2}\) for i=1,…,k, fit the normal random effects model (1) and the proposed model (7) separately.

Step 8

Obtain a posterior median and a 95 percent credible interval of the overall mean from the normal random effects model (1), and those of the overall median from the proposed model (7). Check whether their credible intervals contain the true overall median of 0.000.

Step 9

Obtain a posterior median and a 95 percent credible interval of the I ² from the normal random effects model (1), and those of the ratio of IQR squares from the proposed model (7). Check whether their credible intervals contain the true ratio of IQR squares given by one of either (20.0%, 40.0%, 80.0%).

Step 10

Repeat Steps 1 to Step 9 10,000 times.

Step 11

Using the posterior medians of the overall mean or the overall median obtained in Step 8, compute a bias and a root mean square error around the true overall median of 0.000.

Step 12

Obtain a coverage probability of the overall mean or the overall median by computing the proportion of the time that the 95 percent credible intervals contained the true overall median of 0.000.

Step 13

Using the posterior medians of the I ² or the ratio of IQR squares obtained in Step 9, compute a bias and a root mean square error around the true ratio of IQR squares given by one of either (20.0%, 40.0%, 80.0%).

Step 14

Obtain a coverage probability of the I ² or the ratio of IQR squares by computing the proportion of the time that the 95 percent credible intervals contained the true ratio of IQR squares given by one of either (20.0%, 40.0%, 80.0%).