. Author manuscript; available in PMC: 2015 Nov 10.

Published in final edited form as: Stat Med. 2013 Aug 23;33(2):330–360. doi: 10.1002/sim.5926

Table 2.

Obuchowski-Rockette analysis of Kundel et al [9] data for soft- and hard-copy computed radiographs using trapezoid AUC estimation and jackknife covariance estimation for t = 2 tests, r = 4 readers, c = 95 cases (66 nondiseased, 29 diseased).

Trapezoid AUCs:

Test

1 (Soft-copy) 2 (Hard-copy)

Reader (j) θ̂_1j θ̂_2j

1 0.815 0.854

2 0.767 0.812

3 0.831 0.900

4 0.803 0.798

θ̂_1· = .804 θ̂_2· = .841
ANOVA table:

Source df Sum of squares Mean square

T 1 0.00281054 0.00281054

R 4 0.00715054 0.00238351

T*R 4 0.00140392 0.00046797
Fixed-reader covariance and corresponding correlation estimates computed from jackknife covariance matrix:
${\hat{σ}}_{ε}^{2} = .0022034331, {\hat{Cov}}_{1} = .0011163046, {\hat{Cov}}_{2} = .0.0008438255, {\hat{Cov}}_{3} = .0008871752, {\hat{ρ}}_{1} = 0.507, {\hat{ρ}}_{2} = 0.383, {\hat{ρ}}_{3} = 0.403$
Variance component estimates using Table 1b formulas:
${\hat{σ}}_{R}^{2} = \frac{1}{t} {M S (R) - M S (T * R)} - {\hat{Cov}}_{1} + {\hat{Cov}}_{3} = 0.0007286397$

${\hat{σ}}_{T R}^{2} = M S (T * R) - {\hat{σ}}_{ε}^{2} + {\hat{Cov}}_{1} + max ({\hat{Cov}}_{2} - {\hat{Cov}}_{3}, 0) = - 0.000662504 (typically this would be changed to zero)$
$F_{O R} = \frac{M S (T)}{M S (T * R) + r max ({\hat{Cov}}_{2} - {\hat{Cov}}_{3}, 0)} = 6.00576$
Denominator degrees of freedom:
$d d f_{H} = \frac{{[M S (T * R) + max [r ({\hat{Cov}}_{2} - {\hat{Cov}}_{3}), 0]]}^{2}}{\frac{{[M S (T * R)]}^{2}}{(t - 1) (r - 1)}} = 3$
P -value for H₀: θ₁ = θ₂: p = Pr (F_{(t−1), ddf_H} ≥ F_OR) = .092
95% CI for $θ_{2} - θ_{1} : {\hat{θ}}_{2 \cdot} - {\hat{θ}}_{1 \cdot} \pm t_{d d f_{H}} \sqrt{\frac{2}{r} {M S (T * R) + r max ({\hat{Cov}}_{2} - {\hat{Cov}}_{3}, 0)}} = (- 0.0111940, .086168)$
Single-test 95% confidence intervals based on all of the data. Note: $StdErr = \frac{1}{t r} [M S (R) + (t - 1) M S (T * R) + t r max ({\hat{Cov}}_{2}, 0)]$ .

i θ̂_i StdErr df₂ 95% CI

1 (Soft-copy) 0.804 .0346 46.9 0.734, 0.874

2 (Hard-copy) 0.841 .0346 46.9 0.772, 0.911
Single test 95% confidence intervals using only corresponding test data. Note: ${StdErr}^{(i)} = \sqrt{\frac{1}{r} [M S {(R)}^{(i)} + r * max ({\hat{Cov}}_{2}^{(i)}, 0)]}$ .

i θ̂_i
${\hat{Cov}}_{2}^{(i)}$
MS(R)⁽ⁱ⁾ StdErr⁽ⁱ⁾
$d f_{2}^{(i)}$
95% CI

1 (Soft-copy) 0.804 0.000880 0.000735 0.0326 100.4 0.739, 0.867

2 (Hard-copy) 0.841 0.000808 0.002116 0.0366 19.2 0.765, 0.918