What the pairwise model estimates
Pairwise meta-analysis estimates a summary effect for one intervention-comparator contrast within one outcome and timepoint. Each study contributes an effect estimate y_i and sampling variance v_i. The pooled estimate is a weighted average of the study estimates.
The interpretation depends on the chosen effect measure. For binary outcomes, common measures include odds ratio, risk ratio, and risk difference. For continuous outcomes, common measures include mean difference and standardized mean difference. For time-to-event outcomes, hazard ratios are usually analyzed on the natural log scale.
For common-effect inverse-variance synthesis, weights are w_i = 1 / v_i. The pooled estimate is y_bar = sum(w_i * y_i) / sum(w_i), with standard error SE(y_bar) = sqrt(1 / sum(w_i)).
For random-effects inverse-variance synthesis, weights become w_i = 1 / (v_i + tau^2), where tau^2 is the estimated between-study variance. Recommended settings use REML for tau^2 where the analysis path supports a tau estimator setting; native package-default mode follows the installed package version.
Core extraction fields
- Binary independent-arm studies require events and total participants for each arm.
- Continuous independent-arm studies require mean, standard deviation, and sample size for each arm.
- Time-to-event or adjusted analyses require a hazard ratio or another effect estimate with a standard error, confidence interval, or enough information to reconstruct the standard error.
- Every extracted value should retain the study identifier, outcome name, arm label, timepoint where relevant, source quote, source page, and verification status when available.
- If the study design is not independent-arm randomized data, additional fields are needed before the result should be trusted.
Binary effect-size formulas
For a two-arm binary study, let a be intervention events, b intervention non-events, c comparator events, and d comparator non-events. Let n_e = a + b and n_c = c + d.
Odds ratio: OR = (a * d) / (b * c). The analysis scale is log(OR). Approximate variance is Var(log OR) = 1/a + 1/b + 1/c + 1/d.
Risk ratio: RR = (a / n_e) / (c / n_c). The analysis scale is log(RR). Approximate variance is Var(log RR) = 1/a - 1/n_e + 1/c - 1/n_c.
Risk difference: RD = a / n_e - c / n_c. Approximate variance is Var(RD) = a*b/(n_e^3) + c*d/(n_c^3).
The default binary effect measure is OR. Users can choose OR, RR, or RD. The default binary pooling method is Mantel-Haenszel; Inverse, Peto, and GLMM are available where compatible. Peto and GLMM are restricted to odds-ratio workflows.
Zero cells affect OR and RR calculations. Recommended settings use a 0.5 continuity correction where required and exclude double-zero studies from relative-effect synthesis by default. The double-zero toggle is shown only when double-zero studies exist.
Sparse binary data
Zero cells and rare events require explicit handling. Continuity corrections can make odds ratios or risk ratios computable, but they also influence the estimate. Double-zero studies contain no events in either arm; depending on the selected effect measure and method, they may contribute no information about relative effect while still being relevant for absolute-risk context.
Peto odds ratio, Mantel-Haenszel, inverse-variance, GLMM, and other methods have different assumptions and performance characteristics. The best choice depends on event rarity, arm-size balance, effect magnitude, and study count.
- Report the sparse-data method and continuity correction.
- Run sensitivity analyses when sparse-data decisions can plausibly affect conclusions.
- Check whether double-zero studies were retained, excluded, or used only for descriptive context.
Continuous effect-size formulas
For a two-arm continuous study, let mean_e, sd_e, n_e describe the intervention arm and mean_c, sd_c, n_c describe the comparator arm.
Mean difference: MD = mean_e - mean_c. Approximate variance is Var(MD) = sd_e^2/n_e + sd_c^2/n_c.
Standardized mean difference starts with a pooled standard deviation: s_p = sqrt(((n_e - 1)*sd_e^2 + (n_c - 1)*sd_c^2) / (n_e + n_c - 2)). Cohen d = (mean_e - mean_c) / s_p.
Hedges correction multiplies Cohen d by J = 1 - 3/(4*df - 1), where df = n_e + n_c - 2. Recommended settings use Hedges for SMD. Cohen and Glass are also available.
The approximate variance used for SMD-style preview calculations is (n_e + n_c)/(n_e*n_c) + g^2/(2*(n_e + n_c - 2)), where g is the corrected SMD.
Generic and time-to-event estimates
Generic inverse-variance synthesis uses a reported effect estimate TE and standard error seTE. The model does not recalculate TE from arm-level counts; it uses the reported estimate as supplied, after validation.
For ratio measures such as OR, RR, rate ratio, and HR, TE should be on the natural log scale. A reported HR should usually be entered as log(HR) with SE(log HR), or entered through a workflow that performs that transformation explicitly.
When a 95 percent confidence interval is available for a ratio measure, an approximate standard error can be reconstructed as SE(log effect) = (log(upper) - log(lower)) / (2 * 1.96). For difference measures, SE(effect) = (upper - lower) / (2 * 1.96).
Time-to-event analysis assumes that the reported HR is compatible with a proportional hazards interpretation. The app cannot prove the proportional hazards assumption; reviewers must assess the source analysis.
Heterogeneity and interval settings
Cochran Q is computed as Q = sum(w_i * (y_i - y_bar)^2) on the analysis scale. I^2 is commonly summarized as max(0, (Q - df) / Q) * 100 percent, where df = k - 1.
Random-effects synthesis estimates tau^2, the between-study variance. Recommended settings use REML where supported, while native package-default mode follows the installed package behavior and manual mode allows other tau estimators where supported.
A normal-theory confidence interval for a pooled estimate is y_bar +/- z * SE(y_bar), with z approximately 1.96 for a 95 percent interval. HKSJ/HK intervals can be enabled for supported pairwise random-effects analyses.
A random-effects prediction interval summarizes the expected spread of effects in a similar future setting. Prediction intervals are hidden by default and should be reported only when the model and study count make them meaningful.
Publication checks
- Confirm that every row belongs to the selected outcome, contrast, timepoint, and effect measure.
- Confirm that all ratio measures were analyzed on the natural log scale and back-transformed only for display.
- Confirm that OR, RR, RD, MD, SMD, or HR matches the protocol and clinical interpretation.
- Confirm zero-cell, double-zero, and continuity-correction decisions for sparse binary outcomes.
- Confirm that any adjusted, clustered, paired, matched, crossover, or IPD-derived estimate is entered through generic or design-specific fields rather than treated as ordinary independent-arm data.
- Confirm whether common-effect, random-effects, or both summaries should be interpreted.
- Report the binary method, tau estimator, SMD correction, HKSJ setting, confidence level, continuity correction, and double-zero rule.