On the psychometric function for contrast detection

004226989,‘Xl/O201-02

ON THE PSYCHOMETRIC FUNCTION DETECTION

I5SO2.000

FOR CONTRAST

JACOB NACHMIAS

Department

of Psychology,

University

of Pennsylvania,

Philadelphia,

PA 19104. U.S.A.

(Received 25 February 1980) Abstract-The frequent current use in probability summation calculations of equations of the form. P = 1 - (l-y) exp [- (I/x)u] to represent the psychometric function for contrast detection is based on two assumptions: (I) y can be changed without affecting a and j? (the high-threshold assumption) and (2) fl is the same for all pattern-detecting mechnisms (the homogeneity assumption). Results of yes-no, rating-scale, and forced-choice experiments contradict the high-threshold assumption: estimates of 2 and fl covary with y. Contrary to the homogeneity assumption, bipartite fields yield lower values of b than do 12 c/deg gratings. Some consequences of these findings for probability summation calculations are discussed.

tion, (1) and (2) can be combined

INTRODUCTION

The concept of probability summation, which has a long history of use in visual theorizing,* is playing an increasingly crucial role in current attempts to model spatial and temporal interactions in vision.t The basic idea can be informally stated: if there are n independent “reasons” for an observer reporting the presence of a stimulus, and we denote by Pi the probability of occurrence of any one of these reasons, then the overall probability of reporting the presence of the stimulus, P, is P = 1-

fi (1 - Pi). i= I

P(1) = 1 - exp

-

[

-

[

01 I p a

where r-1 = [XC&“@.

(4)

Equation (4) is also equivalent to vector addition in i-space. hence the title of Quick’s paper, “A vectormagnitude model of contrast detection.” Note that in equation (3) P = 0 when I = 0. However, false alarms do occur, and they have been taken into account by incorporating a quessirlg or false alarm parameter, y, (that is, one of the “reasons” for saying “yes” is actually intensity-independent, with Pi = y). With this final assumption, to be referred to as the lzigll threshold assumption, equation (3) can be rewritten as

(I)

The practical utility of the probability summations concept can be vastly enhanced by using a particular analytic form for the intensity dependence of Pi, that is, the psychometric function : Pi = 1 - exp

to yield

(31 I 81 ;

P(1) = 1 - (1 - y)exp

-

[

where C(~and pi are constants: xi is a threshold parameter, being the value of I for which Pi = 0.63, while pi is a steepness parameter, being the slope of the psychometric function plotted against log I, at I = X. The current use of this equation in vision originates with Quick (1974), though its properties have been discussed previously (Weibull, 1951; Brindley, 1963), and since (Green and Lute, 1975). On the further assumption that all pi are equal, which will be referred to as the homogeneity assump * See, for example, Pirenne, 1943; Peyrou and Piatier. 1946; Lamar et al., 1948; Van den Brink and Bouman, 1954; Brindley, 1954; Blackwell, 1962. t See, for example, Sachs, Nachmias, and Robson, 1971; &anger, 1973; King-Smith and Kulikowski, 1975; Tolburst, 1975; Quick, 1974; Graham, 1977; Legge, 1978; Wilson and Bergen 1979; Watson, 1979. 215

01 I p 31

(5)

This equation usually provides an excellent fit to empirically obtained frequency-of-seeing data from contrast detection experiments. An interesting property of equation (5) is that the shape of the plot of P vs log I solely depends on steepness (/?) and “guessing” (y), and not at all on threshold (a); z merely pegs its location on the abscissa, being the value of I for which P (I) = 0.63 after correction for guessing. This property has several implications for visual psychophysics. Suppose detection of a visual stimulus occurs when the output of at least one of several independent neural mechanisms reaches a criterion level for at least one instant in time. The mechanisms have receptive fields at different locations in the visual field, and these receptive fields may be characterized by different spatial and

216

JACOB

temporal weighting functions. Further, suppose the output of each mechanism is perturbed by noise which varies randomly from moment to moment, and from mechanism to mechanism. On these assumptions, the psychometric function for detecting different types of stimuli must be characterized by the same value of p. even though different values of CI(or even y) might be needed. In other words, when plotted against log I, and corrected for guessing. psychometric functions for detecting any stimulus should have exactly the same shape. This type of shape invariance is to be expected only if the elementary psychometric functions are of the form of equation (2) (Green and Lute, 1975). As stated above, in this scheme of things, variations in the false alarm rate are due only to guessing, and its probability is equal to y. Hence. once 1’ has been taken into account, detection performance for a given type of stimulus should yield identical psychometric functions in forced-choice and yes-no tasks, regardless of the false alarm rate in the latter. That is. the two tasks should lead to the same estimates of threshold (a) and steepness (/I). regardless of the value of the “guessing” parameter i’. Unfortunately, none of these predictions survives empirical tests. The purpose of this note is to document these and other inconvenient facts about psychometric functions in contrast detedction. and to consider some of their consequences. METHODS

The results reported here were obtained over the past several years. usually in the context of specific substantive investigations. For that reason, there is some variation in the type of stimuli and the details of the psychophysical tasks employed. When relevant to the topics under investigation. these differences will be noted. In general, the stimuli were gratings, single bars, and bipartite fields. generated under computer control on the face of a CRT display. Details can be found in Watson (1979) and Watson et al. (1980). Observers viewed the display with both eyes, with natural pupils. (1) Yes-No Several contrast levels of one (or more) type(s) of stimuli were intermixed with about 10% catch trials in each block of up to 200 trials. The observer merely had to indicate after each trial whether or not a stimulus had been detected. (2) Rating-scale Same as yes-no, except the observer had four responses available to indicate degrees of certainty that a stimulus had been presented (or seen). (3) Two-interua/,forced-choice Two temporal intervals were marked by tones on each trial. in one of which a signal was presented. The observer attempted to identify the interval that con-

NACHMIAS

tained the stimulus, and in some experiments, which of two types of stimuli had been presented. In all three methods, presentation of trials was self-paced and feedback was provided after every response. By means of a maximum likelihood procedure based on Chandler’s (1965) STEPIT program (Watson, 1979). equation (5) was fitted separately to the frequency-of-seeing data generated daily by each observer. All three parameters were estimated by this procedure from yes-no and rating-scale data, while only steepness (b) and threshold (9) were estimated from forced-choice data, the “guessing” parameter (7) being set to 0.5 in those cases. RESULTS

Yes-no and forced-choice

e.xperiments

Data from 6 observers and twelve different experimental conditions are available for this comparison. though only a fraction of all possible combinations were run (18 out of 54). For each of these 1II cases, daily estimates of the threshold parameter x and steepness parameter fi were averaged. Experimental conditions differed in regard to types of stimuli presented (stationary sinusoidal gratings of 3, 9 and 12.5 c/deg, gratings of 2 and 8 c/deg drifting at rates corresponding to from I .5 to 12.4 Hz. a bipartite field, a dark bar) and the psychophysical task (detection only of one, or more than one type of stimulus per block of trials, detection and identification of two possible stimuli). Despite the diversity of observers and stimulus conditions, the differences in the parameter estimates from yes-no and forced-choice sessions are very consistent. In all 18 cases. after correction for guessing. yes-no psychometric functions were steeper and had a higher threshold than their forced-choice counterparts. The average ration of estimates is 1.27 for the threshold parameter and 1.40 for the steepness parameter /?. To illustrate these differences. Fig. 1 shows theoretical psychometric functions, after correction for guessing, with the steepness parameter /3 = 4.2 for the yes-no curve and 3.0 for the forced-choice curve. Note that for a “true” detection rate of 0.5, contrasts differing by 2.4 dB are needed in the two procedures. The covariation of parameter estimates reported in this and the following section could have been artifacts of the estimation procedure. This possibility was investigated by running two “Monte Carlo experiments” with assumed guessing parameter (7) of 0.5 and 0.01 respectively. In both “experiments” thresholds (x) were set to 0.1 and steepness parameter (p) to 4.0. The simulated data of 1000 replications of each “experiment” were run through the same estimation procedure which was used to process real data in this study. The resulting mean estimates of ‘1 were 0.0998 and 0.0999 (SD = 0.0330 and 0.0019) and of p were 4.1722 and 4.0307 (SD = 0.660 and 0.3 18). It therefore seems unlikely that estimation artifacts are responsible for the observed dependence of threshold and

On the psychometric

function

for contrast

detection

217

I 0

a

0

0

6

0

4

0

2

0

0

P

-14

-10

-6 I

Fig. 1. Theoretical

psychometric

-2

(DBRE

2

6

II

after correction for guessing, estimated from forced-choice P = 1 - exp[-(1/0.787)3]. Right curve (yes-no): P = 1 - exp - (l/1)4.2.

functions,

and yes-no experiments. Left curve (forced-choice):

steepness parameters on psychophysical task or response criterion. That the standard correction for guessing does not reconcile psychometric functions from yes-no and forced-choice experiments is not a new conclusion. It was stressed nearly two decades ago by the proponents of signal detection theory (see Green and Swets, 1966; Nachmias. 1972). They used it to discredit “high threshold theory” which underlies the standard correction for guessing. In fact, the discrepancy reported here between psychometric functions measured with the two procedures is qualitatively what one would expect if signal detection theory were correct.

I

However, quantitative predictions from signal detection theory depend upon assumptions about the probability distribution of the internal variable generated by repeated presentations of the same stimulus. Curves A and C in Fig. 2 are the yes-no and 24nterval forced-choice psychometric functions (on log abscissa) calculated on the assumption that the distributions are equal-variance Gaussian, with mean proportional to contrast. Curve B is the forced-choice prediction generated by high threshold theory from Curve A. Obviously, Curve B corrected for guessing is Curve A. whereas Curve C similarly corrected would be shallower and displaced to the left.

.o

0.8

0.6

P

0.

2

A 0.0

/!, --

, -14

-10

-6 I

-2

2

6

(DE RE I)

Fig. 2. Curves A and C are theoretical forced-choice and yes-no psychometric functions based on signal detection theory, assuming equal-variance Gaussion distributions, with mean proportional to contrast. Curve B is the high threshold theory predition for forced-choice.

218

JACOB NACHMIAS

Rating-scale

experiments

Results of rating-scale experiments can be used by both signal detection theory and high threshold theory to make quantitative predictions of forced choice performance. In addition. the same results can be used to check whether estimates of the threshold parameter ~1and the steepness parameter fl in equation (5) are really independent of false alarm rate. as they should be on the high threshold theory interpretation of that equation. Observers were asked to use response R, (R,) to indicate high confidence that a stimulus had (had not) been presented and response R2 (R3) to indicate low confidence that a stimulus had (had not) been presented. According to signal detection theory, under these instructions observers set up three simultaneous criterion values along the internal decision axis, and use the 4 responses to report the location of the observation made on each trial relative to the three criteria. On high threshold theory, the 4 responses merely represent (possibly) different propensities to guess. Accordingly. after every session, responses were pooled in three different ways (a) “Yes” = R,, “No” = R, + R3 + Rd. so as to reflect the highest criterion according to signal detection theory. (b) “Yes” = RI + R2, so as to reflect the middle criterion, and (c) “Yes” = RI + R2 + R3. “No” = R, so as to reflect the lowest criterion. For high threshold theory, these 3 ways of grouping should only differ with respect to the underlying composite guessing probability. The 3 sets of daily frequency-of-seeing data obtained in this manner were fit by equation (5) in the usual manner. The relation among the estimates of the threshold and steepness parameters c( and /l to the “guessing” parameter y is summarized in Figs 3 and 4. In Fig. 3, c( is plotted against y on log-log coordinates. For the sake of clarity the data points for each observer are

displaced vertically by an arbitrary amount. Lines connect points based on rating-scale estimates, while the unconnected empty symbols refer to yes-no and forced-choice estimates, when available, for the same observer and stimulus. There is a clear tendency for r to decrease as 7 increases. Recall that (x is the estimated threshold contrast after corrections for guessing. The obtained results indicate that the detection threshold may vary by as much as 2 dB depending on the guessing probability, rather than being a constant. as high threshold theory would have it. Figure 4 shows that for a given observer and stimulus, the steepness parameter p also depends on the “guessing” parameter y rather than being a constant: the greater the presumed guessing probability. the larger the value of fl-’ and hence the shallower the psychometric functions, even after correction for guessing. (The choice of units for the ordinate is justified in the Discussion.) The failure of standard guessing corrections to reconcile psychometric functions based on different response criteria would come as no surprise to signal detection theory. However, as pointed out above, just how these psychometric functions should be related depends, in this theory, on how the parameters of the probability density functions of the internal decision variable vary with stimulus contrast. Conversely, the results of rating-scale experiments provide some constraints on the possible stimulus dependence of the parameters of the underlying functions, and can therefore be used to predict performance in other tasks, such as forced-choice (Green and Swets, 1966). The observer’s rating responses for each contrast level of the stimulus, along with those on catch trials. generate points on an ROC curve. These ROC curves were used in two different ways to predict forcedchoice performance with the same stimulus. (1) Green (1966) has shown theoretically that the area under the

l-

GAMMA

X

l -•

1000

Fig. 3. Estimates of the threshold parameter a vs estimates of the “guessing”

parameter y on log-log coordinates. from rating-scale data (connected points), yes-no data and forced-choice data (empty symbols). Different symbols code different observers or stimulus conditions. Each set of points has been displaced vertically by an arbitrary amount.

219

On the psychometric function for contrast detection

Fig. 4. Estimates of p- ’ vs estimates of log y from rating-scale data (connected points). yes-no data and forced-choice data (empty symbols). Each set of points has been displaced vertically by an arbitrary amount. ROC curve should be equal to the proportion of responses correct in a 2-alternating forced-choice task with the same stimulus. Consequently, areas were calculated from the daily ROC curve obtained by connecting the five available points (including 0, 0 and 1, 1, for each stimulus. (2) By assuming that the underlying distributions are normal, it is possible to *The estimation of parameters of the internal distributions from the rating-scale data was kindly performed by Dr Stanley Klein of the Claremont Colleges. He devised a maximum-likelihood procedure which further assumed that (a) for each observer the several contrast values generated internal distributions whose standard deviations were linear functions of their means, and (b) that the values of these parameters were constant across davs, though the response criteria could vary.

estimate, from an ROC curve, values of the mean and standard deviation of the internal probability distribution corresponding to given stimulus contrasts.* These parameter estimates were, in turn, used to predict proportions of responses correct in the forcedchoice experiment with the same stimulus. Figures 5 and 6 are plots of obtained vs. predicted proportions correct in forced-choice sessions for three observers who detected a 12.5c/deg grating. For Fig. 6, predictions are based on the areas of the ROC curves, while in Fig. 5 they are based on the Gaussian assumption described above. Clearly the area under the rating-scale ROC curve consistently underestimates forced-choice performance for all three observers. On the whole, the Gaussian predictions are

100 -

so-

60-

70-

60-

“O> 60

70

% CORRECT

80

90

I00

(OBTAINED)

Fig. 5. Forced-choice performance predicted from rating-scale data vs forced-choice performance actually obtained. Predictions based on signal detection theory and the assumption of underlying Gaussian distributions. Different symbols code different observers.

220

JACOB NACHMIAS

100

I

VI

I

50

I

60 %

I

I

70 CORRECT

I

I

60

I

I

90

I

I

100

(OBTAINED)

Fig. 6. Forced-choice performance predicted from ratingscale data vs forced-choice performance actually obtained. Predicted values are the areas of rating-scale ROC curves. Different symbols code different observers.

somewhat higher and therefore fit the obtained data better, particularly for one observer (X’s) for whom the Gaussian prediction is extremely successful. The discrepancy between the two predictions is to be expected if the true ROC curve were concave upward: the estimate of the area obtained by connecting five points on the curve would clearly be too small. However, it is doubtful that this factor can entirely explain why the area predictions fall so far short of the observed values. Similarly, it is conceivable but unlikely that the other prediction fails because the form of the underlying distributions depart drastically from Gaussian. There are at least two possible reasons why both procedures might underestimate forced choice performance (I) Observers engage in sub-optimal strategies in rating-scale but not in forced-choice experiments. hence generating points below the “true” ROC curve. If this hypothesis were correct for ratingscale experiments, it would also hold for yes-no experiments. because data from this and other studies (Nachmias. 1968: Emmerich, 1968) indicate that yes-no and rating procedures lead to very similar estimates of the ROC curve. (However, see Markowitz and Swets (1967) for some complications). (2) The “noise” in the two observation intervals of a forcedchoice trial is not independent, as usually assumed. but rather is positively correlated. This would result in better performance in forced-choice tasks than in single interval procedures such as rating-scale or yes-no. Evidence on this possibility is scarce. although Tolhurst (197.5) and Watson (1979) report successful analyses of other types of results which depend on the assumption that noise is independent across time. Srinn~las und ohserrer

differewes

The preceding sections have shown that when equation (5) is fitted to results from yes-no and rating

experiments. the estimates of the threshold and steepness parameters r and /I are correlated with those of the “guessing” parameter 7. Consequently, without equating their false alarm rates, it is meaningless to ask whether yes-no or rating psychometric functions for different stimuli (or different observers) have the same shape. As a practical matter it is easier, in such experiments. to equate false alarm rates at essentially 0 rather than at some moderate value, say, 0.10. Unfortunately, as signal detection theory advocates have pointed out in a somewhat different context (Green and Swets. 1966; Swets, 1964) the higher false alarm rate may be preferable. The reason can be found in Figs 3 and 4 where it appears that log (x and 8-l are nearly linear functions of log 7. This means that log z (and /I-‘) will be changed by the same amount whether ; is changed from 0.001 to 0.01 (values that are hard to discriminate from each other and from 0) or from 0.01 to 0.1 (values that are much easier to discriminate). Nor can the problem of equating the “guessing” parameter be circumvented by concurrently measuring the psychometric functions to be compared. Surely, a single false alarm rate will be obtained. Aside from sampling error, there is still no guarantee the estimates for y will be the same for both stimuli: ; may not merely (or at all) reflect “guessing,” but rather reflect genuine false alarms in the various neural mechanisms. It would seem a far safer procedure to assess stimulus or observer difference in the shape of psychometric functions with data from forced-choice experiments. Here 7 = 0.5, provided the signal occurs equally often in the two intervals. With this method. I have only encountered one stimulus effect which is consistent across observers: gratings produce steeper psychometric functions than do bipartite fields. Results from four observers are summarized’in Table 1. They show that p estimates are consistently higher (by from 2&60’& for gratings than for bipartite fields. The differences, however, are neither large nor significant for any one observer taken alone. The same table also gives some indication of the extent of observer differences in estimates of the steepness parameter /LI.Stimulus conditions were identical for all four observers: overall exposure duration was 500 msec. and contrast was modulated in time by a raised cosine function. Yet there appear to be consistent individual differences in the steepness of psychometric functions. Table I. Estimates of the steepness forced-choice Observer

A.M. R.P. S.S. L.S.

parameter experiments

I2 c ‘deg sinewave 2.50 3.01 2.78 3.37

Bipartite

1.54 2.45 I .49 2.9 1

/? from

field

On the psychometric

function

There are at least two reasons why forced-choice psychometric functions for different stimuli may not have the same shape, even if the high threshold assumptions were correct. (1) The homogeneity assumption is wrong, and the steepness parameter /? is not the same for all neural mechanisms. (2) Equation (2) is not a good enough approximation to the true form of the psychometric function of individual neural mechanism. With any other form, probability summation of n mechanisms results in a psychometric function whose shape depends on the value of n. Hence, if the detection of different stimuli did not involve the same number of mechanisms, the shapes of the psychometric functions might well differ (cf. Wilson and Bergen, 1979). DISCUSSION

The preceding sections demonstrate difficulties with the homogeneity and the high threshold assumptions underlying the frequent usage of equations (4) and (5) for calculating probability summation effects. Contrary to the high threshold assumption, estimates of the steepness parameter /l and the threshold parameter r vary with those of the “guessing” parameter y. and contrary to the homogeneity assumption, all stimuli do not yield the same estimate of p. Do these difficulties invalidate the usual probability summation computations? Not necessarily. Let us consider the consequences of abandoning the high threshold assumption in favor of some form of signal detection theory. Suppose, for a given stimulus the output of each of n mechanisms is a random variable xi (the parameters of the probability distribution of each variable being functions of the stimulus.) In any yes-no trial. the detection of the stimulus

for contrast

221

detection

would depend upon the application of some decision rule on the set of values of xi generated on that trial. A simple (but by no means the only) decision rule is to say “yes” whenever, for a least one mechanism, xi 2 ci (ci = constant criterion). All ci may not be the same, but their values can be altered by instructions. It is still possible that a function of the form of equation (5) could provide a good description of each mechanism’s psychometric function, even though the three parameters in that equation might be interdependent. (In contrast, on the high threshold interpretation of equation (5) for a given mechanism, threshold (zi) depends only on the stimulus, steepness (pi) is constant, and false alarm rate (yi) = 0.) If under a given set of instructions, all steepness (fi) and false alarm (y) parameters were equal, equations (4) and (5) would still describe the probability of criterion being exceeded in at least one mechanism, and hence the overall probability of detecting the stimulus. However, instead of representing the probability of “guessing,” y would be equal to

1 -

i

(1 - yr).

i=l

But although equations (4) and (5) can still be used for probability summation calculations even without the high threshold interpretation, the predicted gain in sensitivity resulting from an increase in the number of contributing neural mechanisms now depends on the criterion value for each mechanism. Suppose the criteria were set in such a way that the steepness parameter (pi) for each mechanism is equal to a constant value (/3). Let us construct an N-component stimulus with the property that (1) each mechanism is sensitive to only one (and a different one) of the components

P

-14

-10

-6 I

(DB

-2

2

6

RE I)

Fig. 7. Right curve: Assumed forced-choice psychometric function for detecting l-component stimulus [P = 1 - 0.5 exp f- 14)]. Left curve: Psychometric function for It-component stimulus. based on highthreshold assumption. Crosses: Psychometric function for 4-component stimulus based on decision rule discussed in text.

222

JACOB NACHMIAS

and (2) for all mechanisms, threshold parameter Xi = 2, a constant. Then from equations (4) and (5). it follows that the observer’s sensitivity to the N-component stimulus, ril is -I = N’&-l

XN

(6)

Since SI- I is also the observer’s sensitivity to any one of the N-components, the ratio of sensitivities to the N- and to a one-component stimulus is N”“, and the db difference in sensitivity is (20/b) log N. Consequently, the expected sensitivity difference for a two component stimulus is 6/b. which is why this unit was chosen for the ordinate in Fig. 4. It can be seen directly from that figure that the amount of expected sensitivity improvement for a 2-component stimulus. due to probability summation alone can be varied by as much as 1 dB by using different criteria. For a 4-component stimulus, criterion variation could produce as much as 2dB difference in the amount of expected improvement due to probability summation-a difference easily discriminable from the OdB predicted by the high threshold assumption. The situation is in some ways more complicated in forced choice experiments. Let us designate by xij the random variable associated with neural mechanism i in observation interval ,j. The observer’s performance in 2-interval, forced-choice experiments depends on the decision rule he applies to the values of xij generated on each trial. There are many possible decision rules. but I have found none under which probability summation calculations reduce exactly to equations (4) and (5). For example, consider one of the simplest decision rules: select the interval that generates the largest value of xi. It can be shown that even if the psychometric function for a I-component stimulus has the form of equation (5) (with the “guessing” parameter ;’ = 0.5). the psychometric function for the N-component stimulus will not be of the same form (see Fig. 7). Compared to the N-component function predicted from the high threshold assumption. this one would be shallower and closer to the one-component function.

CONCLUSIONS (1). The high threshold interpretation of the Weibull psychometric function. which underlies its frequent. current use for probability summation calculations, is plainly wrong. Contrary to that interpretation, threshold (z) and steepness (p) covary with “guessing” (7). rather than being independent of y. (2). With yes-no or rating-scale data, there are at least some other plausible interpretations of the Weibull psychometric function which permit exactly the same kind of probability summation calculations as high threshold theory. However, on these other interpretations, the sensitivity improvement due to probability summation should depend on the number of

stimulus components and the response criterion. The effect of response criterion can be as great as 1 dB for each doubling of the number of stimulus components, (3). With forced-choice data there may be no other interpretation of the Weibull psychometric function which leads to the same probability summation calculations. In fact, under some interpretations, the form of the pyschometric function for multi-component stimuli may differ from that for the l-component stimulus even if the latter is Weibull in form. Ack,lo~/edyenle,lrs-This

work

was Norma comments on an earlier draft of this Brack for improving the intelligibility

grant BNS 75-07658. I thank

supported by NSF Graham for helpful paper. and Jeanette of the present one.

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