Formulas for Within-Study Covariance Matrices and for 1.96-Standard Error Volumes

Thomas A Trikalinos; David C Hoaglin; Kevin M Small; Christopher H Schmid

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

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.

Σ_{k} = (\begin{matrix} σ_{k η}^{2} & 0 \\ σ_{k ξ}^{2} \end{matrix}),

with

σ_{k η}^{2} = \frac{1}{N_{k}^{D} p_{k 1}^{D} (1 - p_{k 1}^{D})},

and

σ_{k ξ}^{2} = \frac{1}{N_{k}^{\bar{D}} p_{k 1}^{\bar{D}} (1 - p_{k 1}^{\bar{D}})},

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:

σ_{k, η m}^{2} = \frac{1}{N_{k}^{D} {\hat{π}}_{k [m : 1]}^{D} (1 - {\hat{π}}_{k [m : 1]}^{D})}

Variance of logit JTPR in study k:

σ_{k, η *}^{2} = \frac{1}{N_{k}^{D} {\hat{π}}_{k, 11}^{D} (1 - {\hat{π}}_{k, 11}^{D})}

Variance for logit FPR in study k for test m:

σ_{k, ξ m}^{2} = \frac{1}{N_{k}^{\bar{D}} {\hat{π}}_{k [m : 1]}^{\bar{D}} (1 - {\hat{π}}_{k [m : 1]}^{\bar{D}})}

Variance for logit JFPR in study k:

σ_{k, ξ *}^{2} = \frac{1}{N_{k}^{\bar{D}} {\hat{π}}_{k, 11}^{\bar{D}} (1 - {\hat{π}}_{k, 11}^{\bar{D}})}

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

σ_{k, η m η t} = \frac{{\hat{π}}_{k [m : 1, t : 1]}^{D} - {\hat{π}}_{k [m : 1]}^{D} {\hat{π}}_{k [t : 1]}^{D}}{N_{k}^{D} {\hat{π}}_{k [m : 1]}^{D} (1 - {\hat{π}}_{k [m : 1]}^{D}) {\hat{π}}_{k [t; 1]}^{D} (1 - {\hat{π}}_{k [t : 1]}^{D})}

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

σ_{k, ξ m ξ t} = \frac{{\hat{π}}_{k [m : 1, t : 1]}^{\bar{D}} - {\hat{π}}_{k [m : 1]}^{\bar{D}} {\hat{π}}_{k [t : 1]}^{\bar{D}}}{N_{k}^{\bar{D}} {\hat{π}}_{k [m : 1]}^{\bar{D}} (1 - {\hat{π}}_{k [m : 1]}^{\bar{D}}) {\hat{π}}_{k [t : 1]}^{\bar{D}} (1 - {\hat{π}}_{k [t : 1]}^{\bar{D}})}

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

σ_{k, η m η *} = \frac{1}{N_{k}^{D} {\hat{π}}_{k [m : 1]}^{D} (1 - {\hat{π}}_{k, 11}^{D})}

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

σ_{k, ξ m ξ *} = \frac{1}{N_{k}^{\bar{D}} {\hat{π}}_{k [m : 1]}^{\bar{D}} (1 - {\hat{π}}_{k, 11}^{\bar{D}})}

Formulas for calculating 1.96-standard-error volumes

Let t₁,…,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...)

\begin{matrix} V_{2} = π t_{1} t_{2} \\ V_{3} = \frac{4}{3} π t_{1} t_{2} t_{3} \\ V_{4} = \frac{1}{2} π^{2} t_{1} t_{2} t_{3} t_{4} \end{matrix}

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,…,λ_M of C. So set $t_{m} = \sqrt{λ_{m}}$ in the formulas above. The 1.96-standard-error volume uncorrected for multiple comparisons is

(z_0.025)^MV_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.

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