Signature
theta_F = sum_(i=1)^k [y_i / v_i] / sum_(i=1)^k [1 / v_i]; SE_F = sqrt(1 / sum_(i=1)^k [1 / v_i])
| Inputs | Definition | Unit |
|---|---|---|
y_i | Effect estimate reported by study i, on a scale where it is approximately normally distributed, such as a log hazard ratio, log odds ratio or mean difference | the chosen effect scale |
v_i | Variance of y_i, the square of its standard error | squared units of y_i |
theta_F | Inverse-variance weighted average of the study estimates under the common-effect model | the scale of y_i, for example a log hazard ratio |
|---|---|---|
SE_F | Standard error of theta_F | the scale of y_i |
kNumber of studies pooled, equal to the length of the y_i and v_i lists (count)
Function
Inverse-variance pooling of aggregate study-level effect estimates
Maps the effect estimate and its variance from each of k studies to one pooled estimate, its standard error and measures of between-study heterogeneity. Each study is weighted by the inverse of the variance of its estimate, so more precise studies count for more. Under a common-effect model the weight uses the within-study variance alone; under a random-effects model it also includes the between-study variance tau2. In health technology assessment the pooled relative effect, usually a log hazard ratio or log odds ratio, is then applied to a baseline from another source in a decision model.
Try this function
Implementations
Excel
Common-effect pooled estimate and standard error from two ranges
With the study estimates in a range named StudyEst and their variances in a range of the same size named StudyVar, the first formula returns theta_F and the second SE_F. SUMPRODUCT evaluates the array division without an array entry.
=SUMPRODUCT(StudyEst/StudyVar)/SUMPRODUCT(1/StudyVar); =SQRT(1/SUMPRODUCT(1/StudyVar))
Assumptions
All studies estimate one common effect
The model is valid when all study estimates target the same underlying effect, which the Cochrane Handbook calls the common-effect assumption. Differences between the y_i are then attributed to sampling error alone.
Estimates on a scale with approximately normal sampling error
Each y_i is approximately normally distributed with a variance v_i that is treated as known. Ratio measures such as hazard ratios and odds ratios are pooled as natural logarithms and transformed back after pooling.
Worked examples
Common-effect pooled log hazard ratio of three illustrative trials
Three illustrative trials report log hazard ratios of -0.60, -0.05 and -0.40 with standard errors of 0.20, 0.10 and 0.25. The weights are 25, 100 and 16, summing to 141, and the weighted estimates sum to -26.4. The pooled log hazard ratio is about -0.1872 with a standard error of about 0.0842, a hazard ratio of 0.829 with a 95% confidence interval of 0.703 to 0.978, as in the article.
k = 3; y_i = [-0.60,-0.05,-0.40]; v_i = [0.04,0.01,0.0625]; theta_F = -0.1872; SE_F = 0.0842
Two equally precise trials pooled by common effect
With equal variances of 0.01 the weights are equal, so the pooled estimate is the simple average of -0.30 and -0.20, and the standard error falls from 0.10 for each trial to about 0.0707.
k = 2; y_i = [-0.30,-0.20]; v_i = [0.01,0.01]; theta_F = -0.25; SE_F = 0.0707
Common errors
Weighting studies by one over the standard error
The inverse-variance weight is 1 over the square of the standard error. Weighting by 1 over the standard error gives weights of 5, 10 and 4 in the three-trial example and a pooled log hazard ratio of about -0.2684 (hazard ratio 0.765) instead of -0.1872 (0.829), because the most precise trial is under-weighted.
Pooling hazard ratios or odds ratios on the ratio scale
Averaging ratio measures directly treats a halving and a doubling of the hazard asymmetrically: two equally weighted hazard ratios of 0.5 and 2.0 average 1.25 on the ratio scale but 1.0 when pooled as logarithms and transformed back. Standard errors derived from a confidence interval also apply to the log scale only.
Sources
Inverse-variance method and common-effect assumption in the Cochrane Handbook
Deeks JJ, Higgins JPT, Altman DG, McKenzie JE, Veroniki AA (editors). Chapter 10: Analysing data and undertaking meta-analyses. In: Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 10.3 (weight equal to the inverse of the variance of the effect estimate), section 10.3.1 (estimate and standard error as the basic data; common-effect assumption) and section 10.3.3 (ratio measures entered as natural logarithms).
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0