The calculations for random effect model meta-analysis are quite straight forward, the followings are steps we need to conduct a random effect model meta analysis using inverse variance weight method.

Random effect meta analysis calculations:

Suppose we got the coefficient($\beta_i$) for a predictor and its variance ($v_i$), $i=1,2,...n$, from each cohort and $n$ is number of cohorts we have.

We assign the weight as inverse of the variance i.e. $w_i=\frac{1}{v_i} \tag{1}$

And the overall coefficient will be calculated as:$\beta=\frac{\sum_{i=1}^{n}w_i\beta_i}{\sum_{i=1}^n w_i}\tag{2}$

The overall variance is $V=\frac{1}{\sum_{i=1}^n w_i}\tag{3}$

The above formulas were for fix effect models, for random effect models we need extra calculations, first we need to calculate $Q$ statistic

$Q=\sum_{i=1}^n w_i(\beta_i-\beta)^2\tag{4}$

Or we can calculate $Q$ statistic as $Q=\sum_{i=1}^n w_i\beta_i^2-\frac{(\sum_{i=1}^nw_i\beta_i)^2}{\sum_{i=1}^nw_i}\tag{5}$

Note formula (4) and (5) are exactly the same, which one you use depending on how you think it is easier to program.

Aftere we get $Q$ statistic we can calculate the $I^2$

$I^2=\begin{cases} &\frac{Q-df}{Q}*100\% \;\;\; \text{ if } Q>df \\ & 0 \;\;\; \text{ if } Q<df \end{cases} \tag{6}$

where $df=\text{(number studies)-1}, i.e. n-1$

For random effect models, we need to calculate the between study variance $\tau^2$

$\tau^2=\begin{cases} &\frac{Q-df}{c} \;\;\; \text{ if } Q>df \\ & 0 \;\;\; \text{ if } Q<df \end{cases} \tag{7}$

and $c=\sum_{i=1}^nw_i-\frac{\sum_{i=1}^nw_i^2}{\sum_{i=1}^nw_i}\tag{8}$

For random effect model, now we need to re-calculate the the variance and weight.

$v_i^*=v_i+\tau^2$ and $w^*=\frac{1}{v_i^*}$, therefore, the overall random effect is calculated as

$\beta^*=\frac{\sum_{i=1}^nw_i^*\beta_i}{\sum_{i=1}^nw_i^*}\tag{9}$

and the overall variance for the overall random effect is calculated as

$V^*=\frac{1}{\sum_{i=1}^kw^*}\tag{10}$

Following the above steps, you can get the exactly same results as the RevMan software for a random effect model meta-analysis.