Signature
t_j = (theta_j - theta_IV) / sqrt(v_j - 1 / W)
| Inputs | Definition | Unit |
|---|---|---|
theta_j | Effect estimate of study j on an approximately normal scale, such as a log hazard ratio, log odds ratio or mean difference | scale of the effect measure |
theta_IV | Common-effect inverse-variance pooled estimate of the K studies, from HE-FM-ADMA-001 | same as theta_j |
v_j | Within-study variance of theta_j, the squared standard error | squared units of theta_j, above zero |
W | Sum of the weights 1/v_k over the K studies, so that 1/W is the variance of the pooled estimate | inverse squared units of theta_j |
t_j | Standardised deviation of study j from the pooled estimate, ranked against the variances | standard deviations of the difference |
|---|
Function
Begg and Mazumdar adjusted rank correlation test function for small-study effects
Maps the effect estimates of K studies in a meta-analysis, on an approximately normal scale such as log hazard ratios, and their within-study variances to the Begg test. Each estimate is standardised against the inverse-variance pooled estimate, and Kendall's rank correlation between the standardised effects and the variances is tested against zero. A significant correlation signals funnel plot asymmetry, a small-study effect, but not its cause: non-reporting of small studies, flawed small trials, genuinely different patients in small trials, artefact and chance can all produce it. The records follow the notation of the Begg Test article. The pooled estimate itself is the common-effect estimate HE-FM-ADMA-001.
Try this function
Implementations
Excel
Standardised Begg effect from named cells
Excel subtracts the named cell holding the pooled estimate from the study's effect and divides by the square root of the study's variance less one over the sum of the weights.
=(Effect-PooledEffect)/SQRT(StudyVariance-1/SumWeights)
Assumptions
Effect estimates on an approximately normal scale for the Begg test
Each theta_j is on a scale where its sampling distribution is close to normal and v_j is its within-study variance, so ratio measures enter on the log scale. Odds ratios and standardised mean differences are naturally correlated with their standard errors, which can create asymmetry without any small-study effect.
Common-effect pooled estimate in the Begg standardisation
The standardisation uses the inverse-variance common-effect estimate, as in Stata's implementation of the test, whatever model the meta-analysis itself reports.
Worked examples
Standardised effect of the most precise of ten trials
In the article's ten illustrative trials the weights sum to 743.66 and the pooled log hazard ratio is -0.1735. Trial 1, with a log hazard ratio of -0.12 and a standard error of 0.06 (variance 0.0036), has a standardised effect of 0.0535 divided by 0.0475, about 1.13.
theta_j = -0.12; theta_IV = -0.1735; v_j = 0.0036; W = 743.66; t_j = 1.1266
Standardised effect of the least precise trial with the largest benefit
Trial 9, with a log hazard ratio of -0.55 and a standard error of 0.28 (variance 0.0784), has a standardised effect of about -1.36, the lowest of the ten, so a large benefit in an imprecise trial drives the negative rank correlation.
theta_j = -0.55; theta_IV = -0.1735; v_j = 0.0784; W = 743.66; t_j = -1.3563
Common errors
Correlating raw effects instead of standardised effects with the variances
Ranking the raw log hazard ratios of the article's trials against their variances gives 10 concordant and 35 discordant pairs, S of -25 and Z of about -2.24 (p about 0.025), a stronger result than the standardised test's -2.06 (computed here for illustration). The raw estimates do not have equal variance, so their ranks mix precision with any small-study effect.
Standardising against a random-effects pooled estimate
The variance of the difference, v_j less 1/W, is derived for the inverse-variance common-effect estimate. Subtracting a random-effects mean, which gives small studies more weight, breaks that variance and the zero sum of weighted deviations, so the standardised values no longer follow the test's definition.
Sources
Stata standardised effect sizes for the Begg rank correlation test
StataCorp. Stata Meta-Analysis Reference Manual: meta bias. College Station, TX: StataCorp LLC; 2025. Methods and formulas, Begg's rank correlation test: the standardised effect size as the difference between the study estimate and the inverse-variance pooled estimate divided by the square root of its variance, the study variance minus the inverse of the sum of the inverse variances.
Cochrane causes of funnel plot asymmetry including correlated effect measures
Page MJ, Higgins JPT, Sterne JAC. Chapter 13: Assessing risk of bias due to missing evidence in a meta-analysis. In: Higgins JPT, Thomas J, Chandler J, et al, editors. Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Table 13.3.b: possible sources of asymmetry, including odds ratios and standardised mean differences being naturally correlated with their standard errors, and chance.
Canonical Identity
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