Abstract

There are three principal alternative assumptions about the form of threshold data collected by presenting each of a number of light intensities repeatedly and determining the probability of discrimination for each. These assumptions are that the data will conform to (a) Poisson sums; (b) normal ogives; or (c) log normal ogives. Decision among these assumed curves is of interest for theoretical reasons, and also to permit selection of a procedure for the analysis of visual threshold data. The present study indicates the similarity among the assumed curves, and reports an analysis of the number of experimental data required to differentiate among them, which number is so large that it appears unlikely that published threshold data are adequate for this purpose. Experimental data are reported which are sufficiently numerous to permit a partial decision among the assumed curves. A total of 27 482 measurements was made by 4 subjects, under constant physical conditions. Data from three of the subjects can be fitted by normal ogives, but not by log normal ogives. Data for the fourth subject can be fitted by log normal ogives, but not by normal ogives. All the data can probably be fitted by one or another Poisson sum, provided there are no theoretical limits on their parameters. That the curves actually represent Poisson sums is regarded as unlikely, however, since a prediction concerning the magnitude of the thresholds is not verified. Our analyses suggest that threshold data can probably be adequately analyzed in terms of either normal or log normal ogives and that visual theories should depend upon other predictions than the form of the threshold data.

© 1953 Optical Society of America

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References

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  1. H. R. Blackwell, J. Opt. Soc. Am. 42, 606 (1952).
    [Crossref] [PubMed]
  2. Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819 (1942).
  3. H. R. Blackwell, Theory and Measurement of Psychophysical Thresholds (to be published).
  4. Brumberg and Vavilov, Bull. de l’Acad. des Sciences de l’URSS, série math. (Akademua Nauk Leningrad, Doklady) 1, 919 (1933).
  5. Reported in Stiles, Proc. Phys. Soc. (London) 56, 329 (1944).
    [Crossref]
  6. Lamar, Hecht, Shlaer, and Hendley, J. Opt. Soc. Am. 38, 741 (1948).
    [Crossref] [PubMed]
  7. Peyrou and Piatier, C. R. de l’Acad. Sci. Paris 223, 589 (1946).
  8. Bouman and van der Velden, J. Opt. Soc. Am. 37, 908 (1947).
    [PubMed]
  9. W. J. Crozier, J. Gen. Physiol. 34, 87 (1950).
  10. H. R. Blackwell, J. Opt. Soc. Am. 36, 624 (1946).
    [Crossref] [PubMed]
  11. Stevens, Morgan, and Volkmann, Am. J. Psychol. 54, 315 (1941).
    [Crossref]
  12. H. R. Blackwell, “Evaluation of the neural quantum theory in terms of visual data,” Am. J. Psychol. (to be published).
  13. See H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton1946).
  14. D. J. Finney, Probit Analysis (Cambridge University Press, Cambridge, 1947).
  15. H. R. Blackwell, Psychophysical Thresholds: Experimental Studies of Methods of Measurement (University of Michigan Press, Ann Arbor, 1953), .

1952 (1)

1950 (1)

W. J. Crozier, J. Gen. Physiol. 34, 87 (1950).

1948 (1)

1947 (1)

1946 (2)

H. R. Blackwell, J. Opt. Soc. Am. 36, 624 (1946).
[Crossref] [PubMed]

Peyrou and Piatier, C. R. de l’Acad. Sci. Paris 223, 589 (1946).

1944 (1)

Reported in Stiles, Proc. Phys. Soc. (London) 56, 329 (1944).
[Crossref]

1942 (1)

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819 (1942).

1941 (1)

Stevens, Morgan, and Volkmann, Am. J. Psychol. 54, 315 (1941).
[Crossref]

Blackwell, H. R.

H. R. Blackwell, J. Opt. Soc. Am. 42, 606 (1952).
[Crossref] [PubMed]

H. R. Blackwell, J. Opt. Soc. Am. 36, 624 (1946).
[Crossref] [PubMed]

H. R. Blackwell, Psychophysical Thresholds: Experimental Studies of Methods of Measurement (University of Michigan Press, Ann Arbor, 1953), .

H. R. Blackwell, “Evaluation of the neural quantum theory in terms of visual data,” Am. J. Psychol. (to be published).

H. R. Blackwell, Theory and Measurement of Psychophysical Thresholds (to be published).

Bouman,

Brumberg,

Brumberg and Vavilov, Bull. de l’Acad. des Sciences de l’URSS, série math. (Akademua Nauk Leningrad, Doklady) 1, 919 (1933).

Cramér, H.

See H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton1946).

Crozier, W. J.

W. J. Crozier, J. Gen. Physiol. 34, 87 (1950).

Finney, D. J.

D. J. Finney, Probit Analysis (Cambridge University Press, Cambridge, 1947).

Hecht,

Lamar, Hecht, Shlaer, and Hendley, J. Opt. Soc. Am. 38, 741 (1948).
[Crossref] [PubMed]

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819 (1942).

Hendley,

Lamar,

Morgan,

Stevens, Morgan, and Volkmann, Am. J. Psychol. 54, 315 (1941).
[Crossref]

Peyrou,

Peyrou and Piatier, C. R. de l’Acad. Sci. Paris 223, 589 (1946).

Piatier,

Peyrou and Piatier, C. R. de l’Acad. Sci. Paris 223, 589 (1946).

Pirenne,

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819 (1942).

Shlaer,

Lamar, Hecht, Shlaer, and Hendley, J. Opt. Soc. Am. 38, 741 (1948).
[Crossref] [PubMed]

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819 (1942).

Stevens,

Stevens, Morgan, and Volkmann, Am. J. Psychol. 54, 315 (1941).
[Crossref]

Stiles,

Reported in Stiles, Proc. Phys. Soc. (London) 56, 329 (1944).
[Crossref]

van der Velden,

Vavilov,

Brumberg and Vavilov, Bull. de l’Acad. des Sciences de l’URSS, série math. (Akademua Nauk Leningrad, Doklady) 1, 919 (1933).

Volkmann,

Stevens, Morgan, and Volkmann, Am. J. Psychol. 54, 315 (1941).
[Crossref]

Am. J. Psychol. (1)

Stevens, Morgan, and Volkmann, Am. J. Psychol. 54, 315 (1941).
[Crossref]

C. R. de l’Acad. Sci. Paris (1)

Peyrou and Piatier, C. R. de l’Acad. Sci. Paris 223, 589 (1946).

J. Gen. Physiol. (2)

W. J. Crozier, J. Gen. Physiol. 34, 87 (1950).

Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819 (1942).

J. Opt. Soc. Am. (4)

Proc. Phys. Soc. (London) (1)

Reported in Stiles, Proc. Phys. Soc. (London) 56, 329 (1944).
[Crossref]

Other (6)

H. R. Blackwell, Theory and Measurement of Psychophysical Thresholds (to be published).

Brumberg and Vavilov, Bull. de l’Acad. des Sciences de l’URSS, série math. (Akademua Nauk Leningrad, Doklady) 1, 919 (1933).

H. R. Blackwell, “Evaluation of the neural quantum theory in terms of visual data,” Am. J. Psychol. (to be published).

See H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton1946).

D. J. Finney, Probit Analysis (Cambridge University Press, Cambridge, 1947).

H. R. Blackwell, Psychophysical Thresholds: Experimental Studies of Methods of Measurement (University of Michigan Press, Ann Arbor, 1953), .

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Figures (9)

Fig. 1
Fig. 1

Theoretical curves. Comparison between a normal ogive (solid curve) and a log normal ogive (dashed curve).

Fig. 2
Fig. 2

Theoretical curves. Comparison between a normal ogive (solid curve) and a simple Poisson sum, with m=5 (dashed curve).

Fig. 3
Fig. 3

Theoretical curves. Comparison between a normal ogive (solid curve) and a special Poisson sum, with m=5 and power of 6 (dashed curve).

Fig. 4
Fig. 4

Data for subject 1. Each point represents 171 target presentations. The solid curve is the normal ogive fitted by the probit analysis.

Fig. 5
Fig. 5

Data for subject 2. Each point represents 171 target presentations. The solid curve is the log normal ogive fitted by the probit analysis. The dashed curve is a normal ogive fitted to the five experimental points of smallest luminance.

Fig. 6
Fig. 6

Data for subject 3. Each point represents 200 target presentations. The solid curve is the normal ogive fitted by the probit analysis.

Fig. 7
Fig. 7

Data for subject 4. Each point represents 100 target presentations. The solid curve is the normal ogive fitted by the probit analysis.

Fig. 8
Fig. 8

Theoretical curves. Comparison between a normal ogive (solid curve) and the curve to be expected if the subject improperly fails to discriminate 5 percent of all target presentations (dashed curve).

Fig. 9
Fig. 9

Theoretical curves. The solid curves are normal ogives; the left has M=0.6, σ=0.2; the right has M=0.9, σ=0.3. The dashed curve was constructed by averaging ordinate values from the solid curves at various abscissa values.

Tables (1)

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Table I Goodness of fit data.

Equations (2)

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p = ( p - C ) / ( 1 - C ) ,
p = p - C ( 1 - C ) ( 1 - ρ ) ,