Appendix AFormulas for Within-Study Covariance Matrices and for 1.96-Standard Error Volumes

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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.

Σk=(σkη20σkξ2),

with

σkη2=1NkDpk1D(1-pk1D),

and

σkξ2=1NkD¯pk1D¯(1-pk1D¯),

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:

σk,ηm2=1NkDπ^k[m:1]D(1-π^k[m:1]D)

Variance of logit JTPR in study k:

σk,η*2=1NkDπ^k,11D(1-π^k,11D)

Variance for logit FPR in study k for test m:

σk,ξm2=1NkD¯π^k[m:1]D¯(1-π^k[m:1]D¯)

Variance for logit JFPR in study k:

σk,ξ*2=1NkD¯π^k,11D¯(1-π^k,11D¯)

Covariance between logit TPRs of tests m and t in study k:

σk,ηmηt=π^k[m:1,t:1]D-π^k[m:1]Dπ^k[t:1]DNkDπ^k[m:1]D(1-π^k[m:1]D)π^k[t;1]D(1-π^k[t:1]D)

Covariance between logit FPRs of tests m and t in study k:

σk,ξmξt=π^k[m:1,t:1]D¯-π^k[m:1]D¯π^k[t:1]D¯NkD¯π^k[m:1]D¯(1-π^k[m:1]D¯)π^k[t:1]D¯(1-π^k[t:1]D¯)

Covariance between logit TPR of test m and logit-JTPR in study k:

σk,ηmη*=1NkDπ^k[m:1]D(1-π^k,11D)

Covariance between logit FPR of test m and logit-JFPR in study k:

σk,ξmξ*=1NkD¯π^k[m:1]D¯(1-π^k,11D¯)

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...)

V2=πt1t2V3=43πt1t2t3V4=12π2t1t2t3t4

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 tm=λm in the formulas above. The 1.96-standard-error volume uncorrected for multiple comparisons is

(z0.025)MVM,

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.