= Solution
Use $w_i=1/v_i$ for the inverse within-study <variances>, and $m=21$. For <Cochran's Q statistic>, the <DerSimonian–Laird estimator> of the between-study <variance> is
$$
\widehat\tau^2=\max\left\{0,\frac{Q-(m-1)}{C}\right\},\qquad
C=\sum_iw_i-\frac{\sum_iw_i^2}{\sum_iw_i}
=400-\frac{16000}{400}=360.
$$
Consequently \b[the estimated heterogeneity variance and standard deviation are]
$$
\boxed{\widehat\tau^2=\frac{38-20}{360}=0.05,\qquad
\widehat\tau=\sqrt{0.05}=0.2236.}
$$
These describe dispersion of underlying study <log odds ratios>, not ordinary sampling error of a single trial. The <between-study heterogeneity> is additional to the within-study <variances>.
The <I-squared statistic> is
$$
\boxed{I^2=\max\left\{0,\frac{Q-(m-1)}Q\right\}\times100\%
=\frac{18}{38}\times100\%=47.37\%.}
$$
It estimates the share of variation beyond that expected from sampling error in this collection of studies. Approximately half the variation is attributed to heterogeneity on that scale; this is neither the percentage of trials with different effects nor a percentage change in mortality. The PDF omits the symbol naming its heterogeneity estimate, so both $\widehat\tau^2$ and $\widehat\tau$ are reported explicitly.
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