Variances for logit-transformed sensitivities and false positive rates for bivariate meta-analysis (single test)
See Table 3 in the main report for notation. The within-study covariance ∑k in study k has zero off-diagonal elements, because sensitivity and specificity are calculated in independent groups.
Variances for logit-transformed sensitivities and false positive rates for the joint multivariate meta-analysis of two tests
Application of the multivariate delta method yields the following formulas. (The notation [m:1] indicates the sum over all patterns in which the outcome on test m is 1).
Variance of logit TPR in study k for test m:
Variance of logit JTPR in study k:
Variance for logit FPR in study k for test m:
Variance for logit JFPR in study k:
Covariance between logit TPRs of tests m and t in study k:
Covariance between logit FPRs of tests m and t in study k:
Covariance between logit TPR of test m and logit-JTPR in study k:
Covariance between logit FPR of test m and logit-JFPR in study k:
Formulas for calculating 1.96-standard-error volumes
Let t1,…,tM be the lengths of the half axes of an ellipsoid of dimension M that corresponds to the contour surface of one standard error. The volume VM included in this one-standard-error surface is calculated by integration. We calculated the first three integrals. (In the following three formulas π = 3.14159...)
For a covariance matrix C we have to calculate the lengths of the half axes for the one-standard-error contour ellipsoid. Rotation to an orthonormal basis automatically provides the lengths of the half axes; these are the square roots of the eigenvalues λ1,…,λM of C. So set in the formulas above. The 1.96-standard-error volume uncorrected for multiple comparisons is
With zα/2 = 1.96 denoting the upper α/2 percentile of the standard normal distribution. For example in the main report, Table 11 the volumes in rows (b) and (c) pertain to four-dimensional models and were calculated using the formula above, for M = 4. The confidence volume in row (a) corresponds to two independent bivariate models. It is calculated as the product of the confidence volumes of dimension 2, one for each bivariate model.
Agency for Healthcare Research and Quality (US), Rockville (MD)
Trikalinos TA, Hoaglin DC, Small KM, et al. Evaluating Practices and Developing Tools for Comparative Effectiveness Reviews of Diagnostic Test Accuracy: Methods for the Joint Meta-Analysis of Multiple Tests [Internet]. Rockville (MD): Agency for Healthcare Research and Quality (US); 2013 Jan. Appendix A, Formulas for Within-Study Covariance Matrices and for 1.96-Standard Error Volumes.