Abstract

In a recent paper [J. Opt. Soc. Am. A 17, 206 (2000)], a modified detectability index, dr, was proposed to accommodate correlations between internal responses in multiple-alternative forced-choice (MAFC) experiments. The derivation given in that work pertained only to two-alternative forced choice, although it was shown empirically that the result held for general MAFC tasks when the correlation between responses is constant. Here we present a rigorous derivation that shows that the dr result generalizes to MAFC tasks in this case.

© 2000 Optical Society of America

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References

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  1. D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).
  2. L. W. Nolte, D. Jaarsma, “More on the detection of one of M orthogonal signals,” J. Acoust. Soc. Am. 41, 497–505 (1967).
    [CrossRef]
  3. D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
    [CrossRef]
  4. P. Elliot, “Forced choice tables,” in Signal Detection and Recognition by Human Observers, J. A. Swets, ed. (Wiley, New York, 1964), pp. 679–684.
  5. M. P. Eckstein, C. K. Abbey, F. O. Bochud, “Visual signal detection in structured backgrounds. IV. Figures of merit for model performance in multiple-alternative forced-choice detection tasks with correlated responses,” J. Opt. Soc. Am. A 17, 206–217 (2000).
    [CrossRef]
  6. A. E. Burgess, X. Li, C. K. Abbey, “Visual signal detectability with two noise components: anomalous masking effects,” J. Opt. Soc. Am. A 14, 2420–2442 (1997).
    [CrossRef]
  7. M. P. Eckstein, A. J. Ahumada, A. B. Watson, “Visual signal detection in structured backgrounds. II. Effects of contrast gain control, background variations, and white noise,” J. Opt. Soc. Am. A 13, 1777–1787 (1997).
    [CrossRef]
  8. In both of the studies referenced, the images have two components of noise. One component is independent across alternatives and across the experimental trials. The second component is replicated (i.e., completely correlated) in each of the alternatives of a single trial but is independent from trial to trial. In this case, the detectability with respect to the independent component is equivalent to the correlation-corrected detectability index proposed by Eckstein et al.5and defined in Eq. (1).
  9. K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, San Diego, Calif., 1979), pp. 36–37.
  10. Ref. 9, pp. 470–473.

2000 (1)

1997 (2)

1978 (1)

D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
[CrossRef]

1967 (1)

L. W. Nolte, D. Jaarsma, “More on the detection of one of M orthogonal signals,” J. Acoust. Soc. Am. 41, 497–505 (1967).
[CrossRef]

Abbey, C. K.

Ahumada, A. J.

Bibby, J. M.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, San Diego, Calif., 1979), pp. 36–37.

Birdsall, T. G.

D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
[CrossRef]

Bochud, F. O.

Burgess, A. E.

Eckstein, M. P.

Elliot, P.

P. Elliot, “Forced choice tables,” in Signal Detection and Recognition by Human Observers, J. A. Swets, ed. (Wiley, New York, 1964), pp. 679–684.

Green, D. M.

D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
[CrossRef]

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

Jaarsma, D.

L. W. Nolte, D. Jaarsma, “More on the detection of one of M orthogonal signals,” J. Acoust. Soc. Am. 41, 497–505 (1967).
[CrossRef]

Kent, J. T.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, San Diego, Calif., 1979), pp. 36–37.

Li, X.

Mardia, K. V.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, San Diego, Calif., 1979), pp. 36–37.

Nolte, L. W.

L. W. Nolte, D. Jaarsma, “More on the detection of one of M orthogonal signals,” J. Acoust. Soc. Am. 41, 497–505 (1967).
[CrossRef]

Swets, J. A.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

Watson, A. B.

J. Acoust. Soc. Am. (1)

L. W. Nolte, D. Jaarsma, “More on the detection of one of M orthogonal signals,” J. Acoust. Soc. Am. 41, 497–505 (1967).
[CrossRef]

J. Opt. Soc. Am. A (3)

Psychol. Rev. (1)

D. M. Green, T. G. Birdsall, “Detection and recognition,” Psychol. Rev. 85, 192–206 (1978).
[CrossRef]

Other (5)

P. Elliot, “Forced choice tables,” in Signal Detection and Recognition by Human Observers, J. A. Swets, ed. (Wiley, New York, 1964), pp. 679–684.

In both of the studies referenced, the images have two components of noise. One component is independent across alternatives and across the experimental trials. The second component is replicated (i.e., completely correlated) in each of the alternatives of a single trial but is independent from trial to trial. In this case, the detectability with respect to the independent component is equivalent to the correlation-corrected detectability index proposed by Eckstein et al.5and defined in Eq. (1).

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, San Diego, Calif., 1979), pp. 36–37.

Ref. 9, pp. 470–473.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

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Equations (28)

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dr=d/1-r.
o=Πm=2Mstep(x1-xm),
PC=dx1dxMfx(x)m=2Mstep(x1-xm)
=-dx1-dx2-dxMfx(x).
fx(x)=1(2π)M/2σMexp-12σ2 (x-μ)T(x-μ).
PC=-dx1-dxM1(2π)M/2σM×exp-12σ2 (x-μ)T(x-μ)×m=2Mstep(x1-xm).
PC=-dz12π-dzM2πexp-12zTz×m=2Mstep(σz1+μ1-σzm-μ0).
step[σ(z1-zm)+μ1-μm]=stepz1-zm+μ1-μ0σ
=step(z1-zm+d),
d=μ1-μ0σ.
PC=-dz12π-dzM2πexp-12zTz×m=2Mstep(z1-zm+d)=-dz12πexp-12 z12Φz1+μ1-μ0σM-1=-dz1ϕ(z1)[Φ(z1+d)]M-1,
fx(x)=1(2π)M/2|Σ|1/2×exp-12 (x-μ)TΣ-1(x-μ),
Σ=σ21rrr1rrr1=σ2C,
PC=dx1dxM1(2π)M/2σM|C|1/2×exp-12σ2 (x-μ)TC-1(x-μ)×m=2Mstep(x1-xm).
PC=dy1dyM1(2π)M/2|C|1/2×exp-12yTC-1y×m=2Mstep(σy1+μ1-σym-μ0).
PC=dy1dyM1(2π)M/2|C|1/2×exp-12yTC-1y×m=2Mstepy1-ym+μ1-μ0σ=dy1dyM1(2π)M/2|C|1/2×exp-12yTC-1y×m=2Mstep(y1-ym+d).
y=C1/2z,
C1/2=abbbabbba,
a=1M [1+(M-1)r+(M-1)1-r],
b=1M [1+(M-1)r-1-r].
ym=azm+bmmzm.
y1-ym=az1+bm1zm-azm-bmmzm
=(a-b)z1-(a-b)zm
=1-r(z1-zm).
C1/2C-1C1/2=I,
dy1dyM=|C|1/2dz1dzM.
PC=dz1  dzM1(2π)M/2exp-12zTz×m=2Mstep[1-r(z1-zm)+d]=dz1  dzM1(2π)M/2exp-12zTz×m=2Mstepz1-zm+d1-r.
PC=-dz1ϕ(z1)-z1+drdz2ϕ(z2)  -z1+drdzMϕ(zM)=-dz1ϕ(z1)Φ(z1+dr)M-1.

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