Algorithm 1.

Sufficient-summary-statistic approach

Step 1: Within-subject analysis

for all Subjects s = 1…S do

Estimate effect size ${\widehat{\theta }}_s$ and its variance ${\widehat{\sigma }}_s^2$

end for

Step 2 : Correlation between effect size and variance

Random effects setting: test $H_0^{\mathrm{\text{corr}}}:{\rho }_{{\theta }_s,{\sigma }_s}=0$

Fixed effect setting: accept $H_0^{\mathrm{\text{corr}}}$

Step 3: Between-subject variance ${\sigma }_{\mathrm{\text{rand}}}^2$

Random effects setting: use, e.g., Equations (41)–(42)

Fixed effect setting: ${\widehat{\sigma }}_{\mathrm{\text{rand}}}^2\leftarrow 0$

Step 4: Population mean effect and variance

for all Subjects s = 1…S do

if $H_0^{\mathrm{\text{corr}}}$ is accepted then

Perform inverse-variance weighting:

${\alpha }_s\leftarrow 1/({\widehat{\sigma }}_s^2+{\widehat{\sigma }}_{\mathrm{\text{rand}}}^2)$

$\widehat{\theta }\leftarrow \sum _{s=1}^S{\alpha }_s{\widehat{\theta }}_s/\sum _{s=1}^S{\alpha }_s$

$\widehat{Var}(\widehat{\theta })\leftarrow 1/\sum _{s=1}^S{\alpha }_s$

else

αs←1/S (equal weighting)

or ${\alpha }_s\leftarrow N_s/\sum _{s=1}^S{N}_s$ (sample-size weighting)

$\widehat{\theta }\leftarrow \sum _{s=1}^S{\alpha }_s{\widehat{\theta }}_s$

$\widehat{Var}(\widehat{\theta })\leftarrow \sum _{s=1}^S{\alpha }_s^2({\widehat{\sigma }}_s^2+{\widehat{\sigma }}_{\mathrm{\text{rand}}}^2)$

end if

end for

Step 5: Statistical inference (H0 : θ = θ0)

$z\leftarrow (\widehat{\theta }-{\theta }_0)/\surd \widehat{Var}(\widehat{\theta })$

z is approximately standard normal distributed

⇒ Reject H0 at 0.05 level if |z| > 1.96