## 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.

with

and

## 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 *t*_{1},…,*t*_{M} 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 *V*_{M} 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 *λ*_{1},…,*λ*_{M} of **C**. So set
${t}_{m}=\sqrt{{\lambda}_{m}}$ in the formulas above. The 1.96-standard-error volume uncorrected for multiple comparisons is

*z*

_{0.025})

^{M}V_{M},

With *z*_{α}_{/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.

## Publication Details

### Copyright

### Publisher

Agency for Healthcare Research and Quality (US), Rockville (MD)

### NLM Citation

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.