Signature
u = gamma_0 + gamma_1 / s; P = Phi_u
| Inputs | Definition | Unit |
|---|---|---|
gamma_0 | Intercept of the Copas probit selection equation; Phi(gamma_0) is the probability of publishing a study with an infinitely large standard error | standard normal scale |
gamma_1 | Coefficient of the precision 1 / s in the Copas probit selection equation, at zero or above | effect scale units, so that gamma_1 / s has none |
s | Reported standard error of the study's effect estimate on the pooling scale, for example a log odds ratio | effect scale units |
Phi_u | Standard normal distribution function at u, entered from NORM.S.DIST(u, TRUE) or pnorm(u) | probability |
u | Index of the probit selection equation, gamma_0 + gamma_1 / s | standard normal scale |
|---|---|---|
P | Marginal probability that a study with standard error s is published, P(Z > 0) | probability |
Function
Copas selection model publication probability and implied number of unpublished studies
Takes the two selection parameters of the Copas selection model, gamma_0 and gamma_1, and the standard errors of the published studies, and returns each study's probability of publication, Phi(gamma_0 + gamma_1 / s), and the number of unpublished studies those parameters imply, M. In a Copas sensitivity analysis gamma_0 and gamma_1 are fixed over a grid, and at each point the mean effect theta, the heterogeneity tau and the correlation rho are estimated by maximising the likelihood of the published estimates conditional on publication. That adjusted estimate has no closed form and comes from software such as the copas function of the R package metasens, so it is not written as a formula here; the publication probability and M are the quantities that label each grid point. The unadjusted estimate the analysis starts from is the common-effect inverse-variance mean of HE-FM-ADMA-001 (random-effects form HE-FM-ADMA-005), and the article's unadjusted 0.3671 is HE-FM-ADMA-001 applied to its twelve illustrative trials. A rank test for small-study effects is HE-FN-BEGG-001, with HE-FM-BEGG-002 and HE-CF-BEGG-001. Carrying an adjusted odds ratio into a decision model uses HE-FM-NMA-004 for the response probability on the new therapy and HE-FM-ICER-001 for the cost per extra responder. Notation follows the Copas Model article.
Try this function
Implementations
Excel
Copas publication probability of one study from named cells
With the selection parameters in cells named SelGammaZero and SelGammaOne and the study's standard error in StudyStdErr, the first formula in a cell named PubIndex returns u and the second in a cell named PubProb returns the publication probability, using NORM.S.DIST with TRUE for the standard normal distribution function.
=SelGammaZero+SelGammaOne/StudyStdErr; =NORM.S.DIST(PubIndex,TRUE)
Assumptions
Probit selection on the reported standard error with fixed selection parameters
Publication depends on the study only through its reported standard error and a standard normal error delta, as in the selection equation Z = gamma_0 + gamma_1 / s + delta that Carpenter, Rücker and Schwarzer state with fixed gamma_0 and gamma_1. In a Copas sensitivity analysis the pair is not estimated from the published studies but fixed over a grid of assumed degrees of selection, because, as Huang and colleagues describe (HE-SR-COPAS-004), the conditional likelihood can peak on a very flat plateau. Dependence on the study's own result enters only through the correlation rho between delta and the sampling error, which this formula does not use.
Precision coefficient at zero or above in the Copas selection equation
gamma_1 is at least zero, so a more precise study is at least as likely to be published as a less precise one; Huang and colleagues describe the term gamma_0 + gamma_1 / s as making studies of high precision more likely to be published (HE-SR-COPAS-004), and Duan and colleagues note that its sign is expected to be positive because larger studies are more likely to be accepted for publication. With gamma_1 equal to zero every study has probability Phi(gamma_0) whatever its size, and with gamma_1 above zero P rises towards 1 as s falls towards zero (derived). The standard error s is above zero.
Standard error on the pooling scale in the Copas selection index
s is the standard error of the estimate on the scale used for pooling, the log odds ratio in the article, so that 1 / s is the precision in the selection equation. A standard error taken on another scale, such as the odds ratio itself, changes the index and the probability.
Worked examples
Copas publication probability of a large trial in the moderate scenario
In the article's moderate scenario, gamma_0 = -1.062 and gamma_1 = 0.2707, a large trial with standard error 0.10 has index 2.707 minus 1.062, which is 1.645, and publication probability 0.95, the value the scenario was built from (illustrative inputs; Phi_u rounded to two decimals, as in the article).
gamma_0 = -1.062; gamma_1 = 0.2707; s = 0.10; u = 1.645; Phi_u = 0.95; P = 0.95
Copas publication probability of a small trial in the moderate scenario
With the same selection parameters a small trial with standard error 0.40 has index 0.67675 minus 1.062, which is -0.38525, and publication probability 0.35, so about two in three such trials would stay unpublished (illustrative inputs).
gamma_0 = -1.062; gamma_1 = 0.2707; s = 0.40; u = -0.38525; Phi_u = 0.35; P = 0.35
Copas publication probability at a very large standard error
At a standard error of 1,000 the precision term is negligible, the index is -1.0617 and the probability is 0.1442, close to Phi(gamma_0) = 0.1441, the limiting case that defines gamma_0 (illustrative).
gamma_0 = -1.062; gamma_1 = 0.2707; s = 1000; u = -1.0617; Phi_u = 0.1442; P = 0.1442
Common errors
Variance used in place of the standard error in the Copas selection index
Writing gamma_1 / s^2 instead of gamma_1 / s inflates the index of every study with a standard error below 1. In the article's moderate scenario the large trial's probability becomes 1.000 instead of 0.95 and the small trial's 0.7356 instead of 0.35, so a scenario meant to leave 65% of small trials unpublished leaves about 26% unpublished (computed here for illustration).
Phi of the Copas intercept read as the probability for the least precise study
Phi(gamma_0) is the limit for an infinitely large standard error, not the probability for the least precise study in the meta-analysis. In the article's moderate scenario Phi(-1.062) is 0.144, while the small trials with standard error 0.40 are published with probability 0.35. The metasens manual calls the cumulative normal of gamma0 approximately the probability that a small study is published, which holds only when gamma_1 / s is close to zero for that study; here it is 0.68 (derived). The copas software reports selection as the probability of publishing the study with the largest standard error, which is Phi(gamma_0 + gamma_1 / s) at that standard error, 0.35 here.
Sources
Carpenter, Rücker and Schwarzer on the Copas probit publication probability
Carpenter J, Rücker G, Schwarzer G. copas: an R package for fitting the Copas selection model. The R Journal. 2009;1(2):31-36 (full text read). Section Copas selection model, equations (2) to (4), states that study i is observed or published if Z_i = gamma_0 + gamma_1 / s_i + delta_i is above zero, with fixed gamma_0 and gamma_1, so that the marginal probability that study i is observed is Pr(Z_i > 0) = Phi(gamma_0 + gamma_1 / s_i), with Phi the standard normal cumulative distribution function. The text interprets Phi(gamma_0) as the marginal probability of publishing a study with infinite standard error and says gamma_1 is associated with the change in publication probability with increasing precision.
metasens reference manual on the Copas selection model and its grid
Schwarzer G, Carpenter JR, Rücker G. metasens, Statistical Methods for Sensitivity Analysis in Meta-Analysis. R package version 1.5-3, 2025 (reference manual read). Function copas, Details, writes the selection model as Z = gamma0 + gamma1 / (SE(i)) + delta(i) and states that the software chooses a grid of gamma0 and gamma1 values corresponding to a range of selection or publication probabilities for the study with the largest treatment effect standard error. The arguments describe gamma0 as the constant in the probit selection model for study publication and gamma1 as the coefficient of study precision (1/standard error).
Duan and colleagues on the roles and sign of the Copas selection parameters
Duan R, Piao J, Marks-Anglin A, Tong J, Lin L, Chu H, Ning J, Chen Y. Testing for publication bias in meta-analysis under Copas selection model. arXiv preprint arXiv:2007.00836. 2020 (full text read). Section 2.1, Copas selection model, states that gamma_0 controls the overall proportion of selection and gamma_1 how selection depends on the size of the study, that the sign of gamma_1 is expected to be positive since larger studies are more likely to be accepted for publication, and gives Pr(Z_i > 0 | s_i) = Phi(gamma_0 + gamma_1 / s_i).
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0