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Binocular interaction and signal detection theory
Vision &s. ~01. 12, pp. 1435-1437. Pcrgamon Press 1972. Printed in Great Britain.
LETTER TO THE EDITORS
BINOCULAR
INTERACTION
AND SIGNAL DETECTION
THEORY
(Received 2 December 197 1)
Gm (1971) argues that experiments comparing binocular with monocular detection should be analysed by the methods of signal detection theory (GREEN.and SWETS,1966). This is a reasonable plea, but the arguments can be taken further than he has done, and they do not necessarily imply that data collected without this theory in mind are valueless. The classical experiments in this field (e.g. PIRENNE,1943) were intended to test, within the framework of a threshold model, whether detection occurred independently for each eye or whether some combination of the monocular signals preceded the threshold mechanism. For independent detection, the binocular detection probability PB is related to the monocular detection probabilities PL and PR by the equation Pf$ = PL + PR _ PLPR
(1)
If the binocular detection rate exceeded that given by such “probability summation”, some degree of binocular combination would be taken to precede the threshold mechanism. This dis~nction has its analogue within a signal detection approach, though that approach admits a wider range of possibilities. Independent monocular decision processes could again be combined by giving an overall “signal” judgment when the decision variable in either channel exceeded criterion (a logical or operation). As Guth points out, equation 1 will then apply not only to the detection probabilities (“hit rates”), but also to the “false alarm rates”. However, it seems misleading to assert, as Guth does, that false alarms “are by no means mistakes” and that in these circumstances “two eyes are no better than either one alone”. Signal detection theory provides us with exactly a method of determining which of two detection performances is “better”, even when both hit and false alarm rates vary; namely the detectability parameter, d’. This parameter can be calculated for binocular detection just as for monocular. MORTON(1967) gives some characteristic theoretical values on the assumption of independent decision processes combined by a logical or; GREEN and SWEE (1966, p. 241) state that the average improvement in d’ on this model is about 0.75 dB, or a factor of l-2 (the actual value varies with the position on the receiver-operating curve at which the decision processes are working). So there is a real sense in which, on this model, two eyes perform better than one. The higher d’ means, for instance, that if the subject adjusted to give the same false alarm rate in binocular as monocular viewing, he would still show a higher hit rate in the binocular case. The particular hit rate obtained depends on the second parameter of the theory, the criterion be&. If the payoffs, i.e. values to the subject, of hits, false alarms, etc., are fixed then a certain setting of the criterion is optimal, in the sense of maximizing expected payoff, Now if the criterion in each monocular channel is set at the optimal value for monocular detection, the observer will not be working at the optimal point for binocular detection; he will be admitting too many false alarms. He can adjust the monocular values ofbeta upwards 1435
1436
LETER TOME EDITORS
to achieve binocular optimality ; this will cut both his false alarm rate and, to a lesser extent, his hit rate. Interestingly, it appears (MORTON,1967; GREENand SWETS,1966) that the same d’ and hit rate are achieved at the optimal setting of a mechanism that takes the logical and of independent monocular decisions. In this case, the monocular criteria are lowered to achieved binocular optimality ; this is because the requirement for detection in both channels increases security against false alarms. From the empirical point of view, the important outcome is that the binocular hit rate given by equation (1) is plausibly a maximum from an ind~~ndent-detection model, For it is the hit rate to be expected if befu is unchan~d in the monocular decision processes; if we suppose that any criterion shift will be towards optimality, the hit rate will be less than this value. Any empirical finding of a hit rate greater than that from equation (1) therefore strongly implies some interaction of the tw,o monocular signals before the decision process. This analysis has assumed the noise in the two channels to be independent. If this is not the case, then the binocular advantage is again reduced; in the extreme case of totally carrelated noise prm both decisions will always be identical, and there will be no binocular advantage at all. If there is some form of binocular interaction preceding the dichotomous de&ion, what form could this take within signal detection theory? One possibility is the “multiple-look” or “integration model” (GREEN and SWETS, 1966, p. 238). In this the decision variables from the two channels are summed. Since the signals add linearly while the random noise, if uncorrelated, is only increased by a factor 42, d’, which is the signal/noise ratio, is increased by a factor 2/2. (This is essentially the model used by CAMPBELLand GREEN (1965) in their comparison of monocular and binocular spatial modulation transfer functions.) Again, if the noise is correlated in the two monocular channels, the binocular advantage will be r&reed. Of course, if the decision variable is ident&d with some physiological variable in the visual pathway, linear addition is only one possible mode of ~m~a~oa of the monocular variables. The model just described assumes that all the noise in the system arises in the separate channels, and summates. However, some or all of the noise might arise after the summation of the monocular variables. If this was true of all the noise, signals would summate but there would be no increase of noise in binocular observation. Again assuming linear s~~tioa, d’ would be inereased by a factor of two. If my of the noise arose after ~rnb~~~on, d would be increased by a factor of more than 42. One posaibk source of such “‘noise” is variability of the criterion (M~RTQN, 1967). Hardly any of the many experiments on monocular vs. binocular &W&on have had this full range of alternatives in mind. However, we may note that any model involving inde~ndent de&ion processes gives the binocular ~van~~ of equation (I), or a lesser advantage, assuming that any criterion shift in the binocular case is in the diz&.ion towards optimality. Thus any grtater binocular hit rate can plausibly be &bed to a decision process acting on a continuous variable resulting from some form of binocular combination. Some recent experiments on binocular detection do show a hit rate greater than that obtained from equation (1) (e.g. COLLIERand ICUEUNSI(Y, 1958; MATfN, 1962; I!%rtrromN, GREENWON, LAppm and C-N, 1966). what is apparently the only binges dragon experiment designed explicitly with signal detection theory in mind (~~~~n, cited in GREENand SWEETS, 1966, p. 247) shows an incX?easein d’ consistent with the last model described above, the linear addition of monocular signals with noise ark& only in the combined pathway. However, there is also published data (e.g. PIRENNE,1943 ; I~~RTLE~
L~R
TO THEEDITORS
1437
and GAG@, 1939; BARANY, 1946; KAHNEMAN, NORMAN and KUBOVY, 1967) which shows a binocular hit rate equal to or less than that from equation (1). Factors can be imagined which would reduce the apparent binocular advantage: the two images might not fall on exactly corresponding points, or the noise in the monocular channels might not be independent. On the other hand, it is difficult to think of any effect which could artefactually inflate the binocular advantage. Thus we may perhaps tentatively conclude that the evidence favours the idea that in binocular detection the decision process acts on a variable that is the result of combining the signals from the two eyes. OLIVER BRADDICK Psychological Laboratory, University of Cambridge, Cambridge, England
BARANY,E.
REFERENCES (1946). A theory of binocular visual acuity and an analysis of the variability of visual acuity.
Acta Ophthal. 24, 63-91. BARTLETT, N. CAMPBELL, F.
R. and GAGNB,R. M. (1939). On binocular summation at threshold. J. exp. Psych& 25,91-99. W. and GREEN,D. G. (1965). Monocular versus binocular visual acuity. Nature, Lord. 208,191. COLLIER, G. and KUBZANSKY, P. (1958). The magnitude of binocular summation as a function of the method of stimulus presentation. J. exp. Psychol. 56,355-361. ERIKSEN,C. W., GREENSPON, T. S., LAPPIN,J. and CARLSON,W. A. (1966). Binocular summation in the perception of form at brief durations. Percept.Psychophys. 1,415419. GREEN,D. M. and S~ETS, J. A. (1966). Signal Detection Theory and Psychophysics, Wiley, New York. Gorn, S. L. (1971). On probability summation. Vision Res. 11, 747-750. KAHNEMAN,D., NORMAN, J. and Kunow, M. (1967). Critical duration for the perception of form: centrally or peripherally determined? J. exp. Psychol. 73, 323-327. MATIN,L. (1962). Binocular summation at the absolute threshold of peripheral vision. J. opt. Sot. Am. 52, 1276-1286. MORTON,J. (1967). Comments on “Interaction of the auditory and visual sensory modalities”. J. acoust. Sot. Am. 42,1342-1343.
P~RENNE, M. H. (1943). Binocular and uniocular thresholds of vision. Nature, Lord. 152, 698.
LETTER TO THE EDITORS
BINOCULAR
INTERACTION
AND SIGNAL DETECTION
THEORY
(Received 2 December 197 1)
Gm (1971) argues that experiments comparing binocular with monocular detection should be analysed by the methods of signal detection theory (GREEN.and SWETS,1966). This is a reasonable plea, but the arguments can be taken further than he has done, and they do not necessarily imply that data collected without this theory in mind are valueless. The classical experiments in this field (e.g. PIRENNE,1943) were intended to test, within the framework of a threshold model, whether detection occurred independently for each eye or whether some combination of the monocular signals preceded the threshold mechanism. For independent detection, the binocular detection probability PB is related to the monocular detection probabilities PL and PR by the equation Pf$ = PL + PR _ PLPR
(1)
If the binocular detection rate exceeded that given by such “probability summation”, some degree of binocular combination would be taken to precede the threshold mechanism. This dis~nction has its analogue within a signal detection approach, though that approach admits a wider range of possibilities. Independent monocular decision processes could again be combined by giving an overall “signal” judgment when the decision variable in either channel exceeded criterion (a logical or operation). As Guth points out, equation 1 will then apply not only to the detection probabilities (“hit rates”), but also to the “false alarm rates”. However, it seems misleading to assert, as Guth does, that false alarms “are by no means mistakes” and that in these circumstances “two eyes are no better than either one alone”. Signal detection theory provides us with exactly a method of determining which of two detection performances is “better”, even when both hit and false alarm rates vary; namely the detectability parameter, d’. This parameter can be calculated for binocular detection just as for monocular. MORTON(1967) gives some characteristic theoretical values on the assumption of independent decision processes combined by a logical or; GREEN and SWEE (1966, p. 241) state that the average improvement in d’ on this model is about 0.75 dB, or a factor of l-2 (the actual value varies with the position on the receiver-operating curve at which the decision processes are working). So there is a real sense in which, on this model, two eyes perform better than one. The higher d’ means, for instance, that if the subject adjusted to give the same false alarm rate in binocular as monocular viewing, he would still show a higher hit rate in the binocular case. The particular hit rate obtained depends on the second parameter of the theory, the criterion be&. If the payoffs, i.e. values to the subject, of hits, false alarms, etc., are fixed then a certain setting of the criterion is optimal, in the sense of maximizing expected payoff, Now if the criterion in each monocular channel is set at the optimal value for monocular detection, the observer will not be working at the optimal point for binocular detection; he will be admitting too many false alarms. He can adjust the monocular values ofbeta upwards 1435
1436
LETER TOME EDITORS
to achieve binocular optimality ; this will cut both his false alarm rate and, to a lesser extent, his hit rate. Interestingly, it appears (MORTON,1967; GREENand SWETS,1966) that the same d’ and hit rate are achieved at the optimal setting of a mechanism that takes the logical and of independent monocular decisions. In this case, the monocular criteria are lowered to achieved binocular optimality ; this is because the requirement for detection in both channels increases security against false alarms. From the empirical point of view, the important outcome is that the binocular hit rate given by equation (1) is plausibly a maximum from an ind~~ndent-detection model, For it is the hit rate to be expected if befu is unchan~d in the monocular decision processes; if we suppose that any criterion shift will be towards optimality, the hit rate will be less than this value. Any empirical finding of a hit rate greater than that from equation (1) therefore strongly implies some interaction of the tw,o monocular signals before the decision process. This analysis has assumed the noise in the two channels to be independent. If this is not the case, then the binocular advantage is again reduced; in the extreme case of totally carrelated noise prm both decisions will always be identical, and there will be no binocular advantage at all. If there is some form of binocular interaction preceding the dichotomous de&ion, what form could this take within signal detection theory? One possibility is the “multiple-look” or “integration model” (GREEN and SWETS, 1966, p. 238). In this the decision variables from the two channels are summed. Since the signals add linearly while the random noise, if uncorrelated, is only increased by a factor 42, d’, which is the signal/noise ratio, is increased by a factor 2/2. (This is essentially the model used by CAMPBELLand GREEN (1965) in their comparison of monocular and binocular spatial modulation transfer functions.) Again, if the noise is correlated in the two monocular channels, the binocular advantage will be r&reed. Of course, if the decision variable is ident&d with some physiological variable in the visual pathway, linear addition is only one possible mode of ~m~a~oa of the monocular variables. The model just described assumes that all the noise in the system arises in the separate channels, and summates. However, some or all of the noise might arise after the summation of the monocular variables. If this was true of all the noise, signals would summate but there would be no increase of noise in binocular observation. Again assuming linear s~~tioa, d’ would be inereased by a factor of two. If my of the noise arose after ~rnb~~~on, d would be increased by a factor of more than 42. One posaibk source of such “‘noise” is variability of the criterion (M~RTQN, 1967). Hardly any of the many experiments on monocular vs. binocular &W&on have had this full range of alternatives in mind. However, we may note that any model involving inde~ndent de&ion processes gives the binocular ~van~~ of equation (I), or a lesser advantage, assuming that any criterion shift in the binocular case is in the diz&.ion towards optimality. Thus any grtater binocular hit rate can plausibly be &bed to a decision process acting on a continuous variable resulting from some form of binocular combination. Some recent experiments on binocular detection do show a hit rate greater than that obtained from equation (1) (e.g. COLLIERand ICUEUNSI(Y, 1958; MATfN, 1962; I!%rtrromN, GREENWON, LAppm and C-N, 1966). what is apparently the only binges dragon experiment designed explicitly with signal detection theory in mind (~~~~n, cited in GREENand SWEETS, 1966, p. 247) shows an incX?easein d’ consistent with the last model described above, the linear addition of monocular signals with noise ark& only in the combined pathway. However, there is also published data (e.g. PIRENNE,1943 ; I~~RTLE~
L~R
TO THEEDITORS
1437
and GAG@, 1939; BARANY, 1946; KAHNEMAN, NORMAN and KUBOVY, 1967) which shows a binocular hit rate equal to or less than that from equation (1). Factors can be imagined which would reduce the apparent binocular advantage: the two images might not fall on exactly corresponding points, or the noise in the monocular channels might not be independent. On the other hand, it is difficult to think of any effect which could artefactually inflate the binocular advantage. Thus we may perhaps tentatively conclude that the evidence favours the idea that in binocular detection the decision process acts on a variable that is the result of combining the signals from the two eyes. OLIVER BRADDICK Psychological Laboratory, University of Cambridge, Cambridge, England
BARANY,E.
REFERENCES (1946). A theory of binocular visual acuity and an analysis of the variability of visual acuity.
Acta Ophthal. 24, 63-91. BARTLETT, N. CAMPBELL, F.
R. and GAGNB,R. M. (1939). On binocular summation at threshold. J. exp. Psych& 25,91-99. W. and GREEN,D. G. (1965). Monocular versus binocular visual acuity. Nature, Lord. 208,191. COLLIER, G. and KUBZANSKY, P. (1958). The magnitude of binocular summation as a function of the method of stimulus presentation. J. exp. Psychol. 56,355-361. ERIKSEN,C. W., GREENSPON, T. S., LAPPIN,J. and CARLSON,W. A. (1966). Binocular summation in the perception of form at brief durations. Percept.Psychophys. 1,415419. GREEN,D. M. and S~ETS, J. A. (1966). Signal Detection Theory and Psychophysics, Wiley, New York. Gorn, S. L. (1971). On probability summation. Vision Res. 11, 747-750. KAHNEMAN,D., NORMAN, J. and Kunow, M. (1967). Critical duration for the perception of form: centrally or peripherally determined? J. exp. Psychol. 73, 323-327. MATIN,L. (1962). Binocular summation at the absolute threshold of peripheral vision. J. opt. Sot. Am. 52, 1276-1286. MORTON,J. (1967). Comments on “Interaction of the auditory and visual sensory modalities”. J. acoust. Sot. Am. 42,1342-1343.
P~RENNE, M. H. (1943). Binocular and uniocular thresholds of vision. Nature, Lord. 152, 698.