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A common-effect synthesis treats study estimates as estimates of one shared effect. For independent approximately unbiased estimates $y_i$ with variances $v_i$, use $\hat{\mu }={\sum }_i{w}_i{y}_i/{\sum }_i{w}_i$ with $w_i=v_i^{-1}$. This model does not accommodate additional between-study heterogeneity.

A fixed-effect meta-analysis models the studies as estimating a common underlying treatment effect. In a generic inverse-variance version, independent log odds ratios $y_i$ have approximate distributions $N(\delta ,v_i)$ and are pooled with weights $1/v_i$. The supplied forest plot instead has the Mantel–Haenszel pooled odds ratio weights: for event/nonevent cells $a_i,b_i,c_i,d_i$ and total $n_i$, the normalized weights are proportional to $b_i{c}_i/n_i$. They reproduce the printed percentages and yield a pooled odds ratio about $2.338$. This is still a common-effect analysis; the two weighting methods should not be silently identified.

A random-effects meta-analysis allows different true effects: $y_i\mid {\delta }_i\sim N({\delta }_i,v_i)$ with ${\delta }_i\sim N(\mu ,{\tau }^2)$. The between-study heterogeneity variance ${\tau }^2$ adds to sampling variance, giving marginal variance $v_i+{\tau }^2$. Its pooled estimate targets the mean of the study-effect distribution rather than one identical effect. Uncertainty about heterogeneity must also be considered.

The overall mean $\mu$ and the between-study heterogeneity $\tau$ encode different information. A proper broad normal distribution prior such as $\mu \sim N(0,2^2)$ is one possible weak prior for the mean log odds ratio; information about the spread of trials alone does not determine its center.

A concrete prior calibration for normal random-effect range can make the factor-of-50 statement simultaneous across all six trials. Conditional on $\tau$, each pair difference has normal distribution ${\beta }_j-{\beta }_k\sim N(0,2{\tau }^2)$. Set $R=\log 50$, $m=\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$, $z={\Phi }^{-1}(1-0.05/(2m))$ andFor every $\tau \le A$, each pair exceeds $R$ in absolute value with probability at most $0.05/m$. The union bound therefore gives $\mathbb{P}({\max }_j{\beta }_j-{\min }_j{\beta }_j>R)\le 0.05$, and integrating over the uniform prior preserves that bound. This is one explicit interpretation of “very unlikely”; a different elicited probability would change the bound. A smoother proper scale prior could be calibrated similarly.

The proposed $1/\tau$ improper prior is unsuitable. The observed-data likelihood function approaches the positive common-effect likelihood as $\tau \downarrow 0$. After restricting $\mu$ and the intercepts to a compact interior region, it is bounded below there by a positive constant. HenceThis is an improper posterior from a log-uniform random-effect scale prior. Proper conditional sampling distributions do not repair the improper joint posterior, and an arbitrary tiny cutoff would make inference depend on that cutoff.

The Student t random-effect model is useful when most studies are comparable but occasional genuine departures are more frequent than a normal distribution hierarchy allows. Its heavier tails permit a study effect far from $\mu$ without forcing a large common between-study heterogeneity scale on every study. In the Gaussian scale mixture representation, a small study-specific ${\lambda }_j$ lowers its precision parameter and weakens its shrinkage.

Use this as robust partial pooling when occasional atypical effects are plausible. Known systematic population or design differences should still be modeled explicitly; a heavy tail cannot identify or correct within-study bias by itself.

Three distinct benefits of a meta-analysis are greater precision, an explicit synthesis of the effect, and assessment of variation across studies.

Combining compatible independent estimates can reduce the standard error, improve statistical power and provide more informative confidence intervals than individual small trials. A prespecified pooled estimate summarises the direction and magnitude of the evidence quantitatively, instead of informally counting statistically significant results. Finally, between-study heterogeneity and subgroup comparisons can reveal whether effects vary with populations or interventions, helping assess generalisability. These benefits require comparable estimands and an appraisal of study validity; pooling cannot repair systematic within-study bias.

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 isConsequently the estimated heterogeneity variance and standard deviation areThese 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 isIt 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 ${\hat{\tau }}^2$ and $\hat{\tau }$ are reported explicitly.

Within-study bias is a systematic displacement of a study's effect estimate from its intended true effect because of its design, conduct, analysis or reporting. Examples include faulty allocation concealment, differential outcome assessment without suitable blinding, informative loss to follow-up, deviations from the intended treatment analysis, and selective reporting of outcomes or analyses. This differs from chance sampling error; the bias need not diminish as a study becomes larger.

It can shift a pooled meta-analysis effect and can create or obscure between-study heterogeneity. A random-effects meta-analysis accommodates dispersion, not systematic invalidity. Publication or non-inclusion of whole studies is a separate selection problem at the synthesis level.

Two useful approaches are risk-of-bias sensitivity analyses and explicit bias adjustment. First, appraise the relevant bias domains and compare the full synthesis with a prespecified synthesis restricted to studies with more credible methods, or stratify by those domains; discuss the resulting loss of precision and possible confounding of study characteristics. Second, if substantive information supports plausible bias magnitudes, use a bias-adjusted meta-analysis with uncertainty about those adjustments, and assess the effect across plausible values. Simply assigning a generic quality score or downweighting a study's sampling variance does not by itself remove its bias.

For independent effect estimates $1.01$ with standard error $0.50$, and $0.99$ with standard error $0.51$, one two-sided Wald test is significant at five percent and the other is not. But their difference is $0.02$ with standard error $\sqrt{0.50^2+0.51^2}\approx 0.714$, giving a negligible test statistic. Hence different significance labels do not establish an interaction or between-study heterogeneity; compare the effects directly.^[ref]