Decision processes in frequency discrimination Wait Jesteadt*

and Sandra

L. Sims

Bioacoustics Laboratory, Eye and Ear Hospital and Schoolof Medicine, Universityof Pittsburgh, Pittsburgh, Pennsylvania15213 (Received 13 December 1974; revised 23 January 1975) The ability to discriminatea change in frequency was measuredin 2IFC, same-different, and yes-no paradigms using two tasks commom in the literature--discrimination of two pure tones differing only in frequency and detection of frequency modulation (FM). In the pure-tone task, differencesin the level of frequency-discriminationperformance as a function of paradigm were similar to those previously observedfor both frequency and intensity discrimination. The level of performance in 2IFC and same-different exceededthe level of performance in yes-no by more than the theory of signal detection would predict. For the FM detection data, differences in the level of performance between paradigms were less than the theory of signal detection would predict. These differences increased,however, when the same subjectswere tested in a task requiring discrimination of two FM tones. These results suggestthat a memory factor is operating in discrimination tasks to a greater degree than in detection tasks. The memory factor is interpreted in terms of criterion variance. Subject Classification: 65.54, 65.35, 65.75.

INTRODUCTION

that the observer

Early theories about the auditory system were primarily concerned with accounting for its frequency-anal-

ysis ability (Ohm, 1843; Helmholtz, 1863). The ability to discriminate small differences in frequency is one obvious measure of the general ability. Although our understanding of the physiological processes underlying frequency resolution has greatly improved in recent

applies the principles

of statistical

decision theory (Wald, 1950)to distributions of sensory events, that the decision axis is monotonic with intensity, and that memory effects are minimal.

Jesteadt and Bilger (1974) recently have reviewed the predictions of the TSD model concerning relative per-

formance in two-interval forced choice (2IFC), samedifferent (SD), and yes-no (YN) paradigms. In the the-

years, our empirical methods for assessing how well

ory of signal detection, the level of performance is ex-

subjects can discriminate small differences in frequency have not. The variability in the reported data is far greater than that in the corresponding intensity-discrimination data. This greater variability can be traced to

pressed in terms

at least two sources: vidual differences

(1) the existence of greater indi-

in frequency-discrimination

ability

than in intensity-discrimination ability (e.g., Jesteadt and Bilger, 1974, Fig. 1), and (2)the lack of consensus concerning the proper procedures to be used in frequency-discrimination tasks as compared to intensity-discrimination tasks. Our primary concern here is with the second source of variability. One aspect of the methodological problem is the continued use of a wide range of psychophysical procedures and data analyses that result in unrelated

measures

of discrimination.

Another

is the continued use of two very different stimulus configurations--pure tones differing in frequency and pure tones that may or may not be frequency modulated.

In recent studies of intensity discrimination, widespread application of the theory of signal detection (TSD)

(Green and Swets, 1966) has led to the use of a limited range of forced-choice paradigms and uniform data analyses. This uniformity reflects the basic assumption that a similar decision process is involved in all psychophysical procedures. The details of the data analyses reflect specific assumptions cuncerning the nature of the decision process. These specific assumptions are testable, in part, because they lead to precise predictions concerning relative performance in the various paradigms. The traditional TSD model of the decision

process (Swets, Tanner, and Birdsall, 1961)assumes 1161

J. Acoust. Soc. Am., Vol. 57, No. 5, May 1975

of the decision

of the difference

distributions

relative

between the means to the standard

deviation of those distributions. The standard prediction is that performance in 2IFC will be higher than perfor-

mancein YN by a factor of V"•. Predictionsconcerning the SD task are less specific because of the ambiguous role played by the standard tone presented in the first interval. The traditional TSD analysis predicts that performance in SD should be poorer than performance in 2IFC by a factor of 2 and poorer than performance in

YN by a factor of V"•. Sorkin(1962)has argued, however, that if the same standard is used on every trial, performance in SD should never be lower than performance

in YN.

For detection, performance has generally been found to differ less across paradigms than TSD would predict,

with the 2IFC/YN performance ratio less than v•- (Swets and Green, 1961; Schulman and Mitchell, 1966; Markowitz and Swets, 1967; Leshowitz, 1969). Subjects apparently do not make perfect use of the additional information present in 2IFC. For intensity discrimination, the differences in performance are greater than TSD

would predict.

The 2IFC/YN performance ratio is

closer to 2 thanto •

(Leshowitz,1969; Viemeister,

1970; Pynn, Braida and Durlach, 1972; Jesteadt and Bilger, 1974). These deviations from the prediction have led some authors to suggest that memory effects play a greater role in discrimination than originally as-

sumed(Viemeister, 1970; Long, 1973; Jesteadt and Bilger, 1974). Why memory should play a greater role in intensity

discrimination

than in detection is not clear.

Copyright ¸ 1975 by the Acoustical Society of America

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Jesteadtand Sims: Decisionprocesses in frequencydiscrimination

I. DECISION PROCESSESIN PURE-TONE MEASURES OF FREQUENCY

DISCRIMINATION

In the few cases where the nature of the decision process has been considered in the frequency-discrimination literature, two very different models have been suggested. One model was proposed by Pierce and Gilbert (1958) to account for differences in results obtained with

AB and ABX paradigms (Rosenblith and Stevens, 1953). It closely parallels the traditional that the decision

axis

is assumed

TSD analysis, except to be monotonic

with

frequency rather than intensity. This model predicts the relations among paradigms outlined above. The

other model, proposedby Wickelgren (1969), deals only with the SD paradigm and assumes that judgments of

"sameness" are based on the familiarity of the comparison tone.

The decision

axis is monotonic

with familiar-

ity, not frequency. The subject could not use such a system to make high-low judgments and Wickelgren argues that these judgments are mediated by a separate process. Wickelgren's model, therefore, predicts that performance in 2IFC is unrelated to performance in SD. Such a weak prediction would be of little interest were it not for Wickelgren's prediction that performance in SD is generally better than performance in 2IFC in frequency-discrimination tasks. Since the superiority of performance in 2IFC in intensity discrimination has been well established, Wickelgren is, in fact, asserting that there are basic differences in the processing of intensity and frequency information. A number of other authors have made similar assertions (e.g., Stevens, 1957; Tanner and Rivette, 1964; Whipple, 1901). Because the models proposed by Pierce and Gilbert (1958) and by Wickelgren (1969) make differential predictions concerning relative performance in different paradigms, the most straightforward test of the models is to compare performance in these paradigms. Jesteadt

and Bilger (1974) found the relations among 2IFC, SD, and YN paradigms in frequency discrimination to be very similar to those in intensity discrimination and suggested that the same decision process was operating in both cases. The fact that performance in 2IFC greatly exceeded performance in SD and YN led them to re-

ject Wickelgren's model. As in other paradigm comparison studies involving discrimination rather than detection, however, they found it necessary to assume the existence of a memory factor to account for the large deviations from the TSD predictions. Thus, neither model of the decision process was supported unequivocally. Variability in the performance ratios for individual subjects and relatively poor overall performance in the frequency-discrimination tasks increased the ambiguity. One purpose of the present study was to replicate the fixed-standard, frequency-discrimination portion of the earlier

study.

1162

Shower and Biddulph than to Riesz. These references often iuvolve a comparison of new pure-tone data to the earlier FM data, with the implicit assumption that the two stimulus configurations measure the same properties

of the auditory system (e.g., Henning, 1966). We are aware of no direct evidence to support this assumption, and there is, in fact, some evidence to the contrary. In general, studies using pure-tone stimuli have reported smaller DL's than those reported by Shower and Biddulph,

with especially large differences below 500 Hz (Stevens, 1954). Nordmark (1968) has argued that the data reported by Shower and Biddulph are not representative

and that data from more recent FM studies (Meurman, 1954; Groen and Versteegh, 1957; Filling, 1958), when plotted correctly, show DL's as a function of frequency similar to those obtained in studies using pure tones.

The pattern described by Stevens (1954) can still be observed, however, in the replotted data. There are no data in the literature concerning decision processes in FM measures of frequency discrimination. Knowledge of these processes should increase our understanding of the data collected with this stimulus configuration and its relation to pure-tone data. FM conditions were therefore included in the present study to permit a direct comparison of the decision processes in FM and pure-tone measures of discrimination. Because the difference between these measures is greatest at low frequencies, discrimination was tested at 250 Hz as well as at 1000 Hz in all paradigms. III.

A.

EXPERIMENT

I

Method

1. Sub/ects Three members of the laboratory staff served as subjects. They all had prior experience as listeners in frequency-discrimination tasks. 2.

Conditions

Discrimination was tested at 250 and 1000 Hz using pure-tone and FM signals in each of three psychophysical paradigms. All signals were presented at 60 dB SPL. In the pure-tone-signal conditions, the subjects' task was to discriminate between two possible pure tones differing only in frequency. One of the two possible tones was the 250-

or 1000-Hz

standard

and the other

was

higher in frequency by some amount. In the FM-signal conditions, the subjects' task was to discriminate between a 250- or 1000-Hz pure-tone and an FM signal with the same center frequency and an 8-Hz modulation rate.

of both frequency discrimination (Shower and Biddulph, 1931) and intensity discrimination (Riesz, 1928), there

In each of the four conditions (pure-tone and FM signals at 250 or 1000 Hz), discrimination was tested at three degrees of difficulty to generate psychometric functions. In the pure-tone-signal conditions, different degrees of difficulty were obtained by varying the difference in frequency between the standard and the other possible tone. In the FM signal conditions, they were obtained

are many more references

by varying the range of the frequency modulation.

II.

DECISION

FREQUENCY While

PROCESSES

IN FM MEASURES

OF

DISCRIMINATION

modulated

tones

were

used in the classic

in the current

literature

studies

to

J. Acoust. Soc. Am., Vol. 57, No. 5, May 1975

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JesteadtandSims: Decisionprocesses in frequencydiscrimination

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3. Paradigms

In the 2IFC paradigm, the subjects were instructed to listen to two signals and decide in which interval the higher of the two possible frequencies or the FM signal occurred. In the SD paradigm, the subjects were in-

structed to listen to two signals and vote "Different" if the second signal was the higher frequency or was modulated. In the YN paradigm, the subjects were instruct-

ed to listen to one signal and vote "Yes" if it was the higher frequency or was modulated. 4. Apparatus The signals for the pure-tone

conditions were gener-

ated by a programmable oscillator (Krohn-Hite 453-2). The signals for the FM conditions were generated by a

Wavetek oscillator (model 116). A second Wavetek oscillator was used to generate the 8-Hz sinusoidal modu-

lating waveform. A programmable attenuator (Wolf, 1972) controlled the amplitude of the modulating wave-

form and, therefore, the rangeof the modulation. • The

I

2

3

I

AF

pure-tone and FM signals were gated by an electronic

IN

2

3

4

5

6

Hz

switch (Grason Stadler model 829D)with a 25-msec

FIG. 1.

rise-decay

digrns--2IFC (squares), same-different (diamonds), and yesno (circles). The six panels contain data for 250 Hz (on the left) and 1000 Hz for the three subjects, SLS, LJF, and CMR.

time.

These devices were controlled by a

PDP-15/20 computer that was also used to record subjects' responses. The softwave and interface have been

Pure-tone frequency discrimination

in three para-

described by Jesteadt (1972). The signals were presented monaurally through TDH-49 phones to the three subjects in separate sound-treated

with the assumptions of the TSD model and Wickelgren's

rooms.

(1969) model of the decision process. We refrain from 5.

Procedure

labeling this measure of performance d • because that

A 2IFC or SD trial consisted of a warning interval, two 500-msec observation intervals separated by 500 msec, an answer interval, and a correct-answer feedback interval. The same pattern was used for YN trials, with the exception of the intersignal interval and second observation

In each daily 2-h session, 100 trials were run at each

preceded by six days of practice during which the appropriate pure-tone frequency differences ranges were determined.

and FM

A preliminary analysis of the data indicated that discrimination performance in the FM conditions was much lower than that in the pure-tone conditions. The FM conditions were therefore repeated using larger FM ranges, so that the level of difficulty of the pure-tone and FM conditions would be more nearly equal. The largest of the three FM ranges in the original data collection was used as the smallest of the three FM ranges in the replication.

tion.

We refer

to the uncorre.cted

discrimination

mea-

sures for different paradigms as d•.zFc, dsD, or these measures for different paradigms, we use the term

The values of dp for the PT and FM conditions are presented in Figs. 1 and 2, respectively. Each panel contains performance data for an individual subject for

the three paradigms. For the FM conditions, the d• for the middle FM range represents the mean of the d• values for the original data collection and the replication. Wherever performance by a given subject for the larger FM ranges was too high to measure, data for those FM ranges were discarded for all three paradigms. The best fitting linear psychometric function has been drawn through each set of points representing performance in a given paradigm, with the restriction that all three

functions (for each subject) pass through a commonx intercept. Our reasons for imposing this restriction are the same as those cited by Jesteadt and Bilger (1974) for fitting two-point functions with straight lines through the origin.

It is the minimum

restriction

that allows us

to compare performance in different paradigms for a given subject and condition in terms of one parameter

Results

d•/Hz, and in a few cases it reducesthe effect of what

For each level of difficulty, the proportions correct for each of the two possible responses were converted to z scores.

pensate for the difference in performance between paradigms predicted by TSD. We have made no such correc-

Where a general label is needed for more than one of

interval.

of three points on the psychometric function for each of the three paradigms for a given condition (pure-tone or FM signal at 250 or 1000 Hz). A total of 200 trials for each point on each function was obtained by running each of the four conditions on two different days in a counterbalanced order. The eight days of data collection were

B.

term implies that a V• correction has been made to com-

The two z scores

a measure of discrimination

were

then

performance

added to obtain

appears to be poor estimates of performance. The more severe restriction that the functions pass through the origin obviously could not be applied in the FM condi-

consistent

tions.

J. Acoust. Soc. Am., Vol. 57, No. 5, May 1975

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Jesteadtand Sims: Decisionprocesses in frequencydiscrimination

1164

TABLE II. Ratios of the slopes of psychometric functions describing relative frequency-discrimination performance in the different paradigms for each condition in Experiment 1.

4 I I I I IID// I I0It 3 ./•

./

2

Slope ratio

//

Condition

Subject

2IFC/YN

2IFC/SD

SD/YN

PT-250

1

1.98

1.57

1.26

2

2.40

1.72

1.40

3

2.41

1.51

1.59

i .

i

I

i

I

i

i

i

I

_

_

_

_

_

_

PT-1000

_

_

Mean

2.26

1.60

1.42

1

2.57

2.63

0.98

2

2.62

1.42

1.85

3

2.69

'

2.39

1.12

Mean

2.65

2.16

1.32

1

1.47

1.54

0.95

2

0.94

1.00

0.94

3

1.56

1.39

1.12

Mean

1.32

1.31

1.00

_

_

FM-250 i

i

i

i

i

i

i

i

/

2 •,•i'""'

FM-1000

1'

0.96

0.81

1.19

2

1.08

1.08

1.00

3

1.28

1.11

1.16

Mean

1.11

1.00

1.12

_

i ?

4.

i 8

6

FM

RANGE

IN Hz

FIG. 2. FM detection in three paradigrns--2IFC (squares), same-different (diamonds), and yes-no (circles). The six panels contain data for 250 Hz (on the left) and 1000 Hz for the

described in terms of the ratio of the slopes of the psychometric functions. The slope ratio provides an esti-

mate of the ratio of the values of d• that would be ob-

three subjects.

tained

We briefly consider overall differences between conditions before dealing with relations between paradigms. It is obvious from a comparison of Figs. 1 and 2 that performance was generally poorer in the FM conditions than in the pure-tone conditions. While the slopes of the psychometric functions for the FM conditions are generally not as steep as those for the pure-tone conditions, the major difference is that the x-axis intercepts for the FM conditions are markedly greater than zero. Implications of this difference are discussed below.

if the same

increment

size

were

used in both

paradigms, and has the advantage of being based on performance for several increment sizes. Slope ratios for the three possible pairwise comparisons of paradigms are presented in Table II. Only two of these, of course, contain independent information. The third is provided for convenience.

The ratios for the pure-tone conditions in Table II agree with the earlier finding by Jesteadt and Bilger

(1974) that discrimination perœormanceis consistently better in the 2IFC paradigm than in the other two paradigms by factors greater than those predicted by the

To facilitate comparisons across conditions and comparisons with other studies, we used the functions in

traditional TSD analysis, a difference in the opposite direction from that predicted by Wickelgren (1969).

Figs. 1 and 2 to estimate the values of Af/f required for 75% correct performance in 2IFC. These values are presented in Table I. The difference in zXf/f be-

The FM performance ratios are markedly lower than those for the pure-tone conditions. In three out of six

tween the FM and pure-tone conditions is clearly larger at 250 than at 1000 Hz.

The data are therefore

in better

agreement with the. summary presented by Stevens

(1954) than with the summary presented by Nordmark (•968). The relations between any two paradigms can best be

cases, the 2IFC/YN ratio obtained in the FM conditions is as large or larger than 2IFC/YN ratios obtained for detection of pure tones in noise (see Jesteadt and Bilger, 1974, Table I). In the other three cases, performance appears to be uniform across paradigms.

Superficially, at least, the performance ratios for the pure-tone data resemble those for intensity discrimination, while the ratios for the FM data resemble those for detection.

TABLE I.

1

formation but rather to a general difference between dis-

2

3

4

PT-250

0. 0037

0. 0034

0. 0030

0. 0034

0. 0019

0. 0021

0. 0020

0. 0020

FM-250

0. 0166

0. 0245

0. 0121

0. 0177

0. 0098

as

Subject

PT-1000

FM-1000

can be described

1.

Values of Af/f required for 75% correct in 2IFC

for each condition in Experiment

Condition

Since the FM task

detection of modulation, it may be that the difference in performance ratios we obtained is not related specifically to a difference in processing pure-tone and FM in-

0. 0064

0. 0076

0. 0079

crimination and detection tasks. To test this hypothesis, we conducted a second experiment to measure performance in the 2IFC, SD, and YN paradigms in an FM discrimination

task in which the standard

as well

as the

comparison signal was modulated and subjects were asked to discriminate between the two modulation ranges.

J. Acoust. Soc. Am., Vol. 57, No. 5, May 1975

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1165 IV. A.

Jesteadtand Sims: Decisionprocesses in frequencydiscrimination EXPERIMENT

2

1165

TABLE III. Ratios of the slopes of psychometric functions describing relative frequency-discrimination performance in the different paradigms in Experiment 2.

Method

The subjects used in Experiment 1 served as subjects in this experiment. The standard was an FM signal øø Hz range centered on • 000 Hz. T h• modulated over a o•rate of modulation was 8 Hz. The comparison signals

Slope ratio

Subject

2IFC/YN

1

1.67

1.73

0.96

2

2.00

1.54

1.30

3

1.38

1.29

1.07

Mean

1.68

1.52

1.11

were modulated at the same rate over larger ranges and four different comparison ranges were used to obtain four-point psychometric functions for each paradigm. Because individual differences in this task were greater than those encountered in Experiment 1, we used different comparison ranges for each observer. In all other details, Experiment 2 was identical to the FM portion of Experiment 1.

V.

DISCUSSION

B.

A.

Overall performance

Results

The data analysis was also identical to that in Experi-

ment 1. Values of dpare presentedin Fig. 3. The values of Af on the abcissa indicate the differences between the comparison ranges of modulation and the 32-Hz standard range. Table III contains the ratios of the slopes of functions fitted to the data in Fig. 3. The new psychometric

functions have intercepts

closer

to zero and shallower slopes, especially in the SD and YN paradigms.

A comparison of Tables II and III in-

dicates that the 2IFC/SD and the 2IFC/YN ratios are larger for FM discrimination

than for FM detection.

The SD/YN performance ratio showslittle change. While the new performance ratios are not as large as those for pure-tone discrimination, they are larger than both the TSD predictions and the ratios typically obtained

2IFC/SD

SD/YN

for detection of pure tones in noise.

There are several studies in which pure-tone frequency discrimination has been measured using a 2IFC paradigm with signal durations and levels comparable

those in the current study (Harris, 1952; Henning, 1970; Moore, 1973). The variability in the data obtained with 2IFC is much smaller than the variability observed in the frequency-discrimination literature as a whole. These studies, therefore, provide a good benchmark against which to compare new data. The mean Af/f at 250 Hz for 2IFC in the present study (Table I) is 0. 0034. The mean values across subjects for the ear-

lier studies are 0.0031, 0. 0027, and 0. 0028, respec-

tively. The mean Af/f at 1000Hz in Table II is 0. 0020, while the corresponding means for the earlier are 0. 0013, 0.0014, and 0.0018. The quality of the FM data is more difficult

studies to evaluate.

We are not aware of any FM studies that have used procedures comparable to those in the present study. The

values of Af/f for the FM conditions in Table I are larger than those in the literature, but all of the previously published values were obtained with the method

of adjustment.

We know that pure-tone measures of

Af/f obtainedwiththe methodof adjustment(Nordmark, 1968; Rakowski, 1971) are generally smaller than those obtainedwith 2IFC. The difference in procedure, therefore, may accountfor the larger values of Af/f we observed. •' The relation between discrimination at 250 and 1000 Hz for the FM conditions

in Table I is similar

to that observed in previous FM studies. The relatively poor performance in detecting frequency modulation at low center frequencies is apparently not a function of procedure.

There is general agreement across all of the studies in the literature that pure-tone procedures result in

smaller values of Af/f than FM procedures.

The data

in the present study indicate that an important factor contributing to this difference is the nonzero intercept of the psychometric functions for FM detection. No matter what scales were used in plotting the FM data for Experiment 1, at least two of our three subjects would show chance performance for the smallest FM range.

FIG. 3. FM discrimination in three paradigms--2IFC

(squares), same-different (diamonds),and yes-no (circles). The three panels containdata for the three subjects.

The large difference in intercepts for the pure-tone and FM psychometric functions may give an indication

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1166

of how subjects are performing the FM task. If the subjects perform both the pure-tone and the FM tasks by timing the periods of individual cycles, as Nordmark (1968) has suggested, then we would expect to see psychometric

functions

in both

tasks

with

a common

inter-

cept near zero and a difference only in slope. The fact that this is not the case may indicate that the subjects are performing a spectral analysis of the FM signal,

as Kock (1937) and Feth, Wolf, and Bilger (1969) have suggested.

B. Relations between paradigms The primary goals of the present study were to replicate one condition of the study reported by Jesteadt

and Bilger (1974) and to extend the range of conditions to include low-frequency signals and FM. The performance ratios for the replicated condition, pure-tone

frequency discrimination at 1000 Hz, are in agreement with

1166

Jesteadt and Sims: Decision processesin frequency discrimination

the earlier

data that

indicate

that

the relations

be-

tween paradigms are similar to those for intensity discrimination. The pure-tone data for 250 Hz show the same relations as those for 1000 Hz, within the limits of the variability always observed in performance ratios. 3

The pure-tone data do not support Wickelgren's (1969) model of pitch memory and decision processes in the same-different paradigm. The upper and lower limits of performance in SD can always be specified in terms

of performance in 2IFC or YN.

There are, therefore,

no effects unique to $D that require an independent model for that paradigm. The data do support Wickelgren's assertion that memory factors play an important role in frequency-discrimination tasks. The most straightforward way of accounting for the pure-tone data is to assume that the TSD model of the decision process is correct, but that memory for the standard is far from perfect. If the memory factor is considered as an additional source of decision variance and some assumptions are made concerning its magnitude relative to the variance in a single observation, specific predictions can again be made concerning relations between paradigms. Vie-

meister (1970) has discussed the special case where memory variance

and observation variance

are equal.

In this case, the 2IFC/YN performance ratio shouldbe 2.0, while the SD/YN ratio shouldremain 1.0. In the SD paradigm, the decision variance is the same whether the observer

listens

to the first

interval

or relies

on

-

memory for the standard as he would in YN. In general, as memory variance increases from zero to some amount

equal to the observationvariance, the 2IFC/YN and 2IFC/SD ratios shouldincreasefrom the factor of • predicted by TSD to the factor of 2.0 predicted by Viemeister (1970). As memory variance increases beyond

observationvariance, the 2IFC/YN and SD/YN ratios shouldincrease indefinitely, while the 2IFC/SD ratio remains constant at 2.0.

For memory variance in this

range, the subject should listen to the standard in SD, rather than relying on memory. The subject should then be performing the same operations in SD and 2IFC, and the relation between these two paradigms should be governed by the use of the order information in 2IFC,

not by memory for the standard.

When the SD/YN ratio is significantly greater than 1.0, it can be used to estimate the magnitude of the memory variance

in YN relative

to the sensation variance

in a singleobservation.4 Usingthe meanSD/YN ratio for both pure-tone conditions, 1.37, we estimate that the memory variance exceeds the observation variance in YN by a factor of 2.75. If we were to assume that memory variance plays some role in SD as well as YN, our estimate of the relative magnitude of the memory variance would increase. The pure-tone data are therefore consistent with the assumption that memory variance not only exists, but is greater than the sensory variance in a single observation. A possible interpretation of the memory factor is discussed below. The relations between paradigms observed in the FM conditions indicate still another way in which standard pure-tone and FM measures differ from one another. While the pure-tone conditions showed paradigm effects much larger than predicted on the basis of TSD, the FM conditions showed effects that were smaller than predicted

and in some

cases

nonexistent.

As we noted

above, paradigm comparison studies involving detection of pure tones also have reported effects smaller than the TSD predictions.

The range of paradigm effects observed in the two experiments indicates that there are no fixed values for the ratios describing relative performance in different paradigms. The results suggest that at least two factors govern the size of the performance ratios. One of these is memory variance. For the most part, the pattern of performance ratios in the FM detection, FM discrimination, and pure-tone discrimination conditions can be accounted for by assuming that these tasks fall at different points on a memory-variance continuum. The small paradigm effects observed in the FM detection conditions suggest that memory for the standard is essentially perfect. In the FM discrimination condition, memory variance appears to be present, but is less than the observation variance. In the pure-tone discrimination conditions, as we noted above, memory variance appears to be greater than observation variance. The memory-variance assumption cannot account for the fact that in the FM detection task, and in the detec-

tion of pure-tones in noise, the 2IFC/YN and 2IFC/SD performance ratios are less than TSD would predict. The prediction that performance is better in 2IFC than in SD or YN reflects the assumption that the decision is based

on the difference

in the first

between

and second intervals.

the observations

In 2IFC

made

this difference

can be either positive or negative, while in SD or YN (where memory for the standard replaces one of the intervals) it can only be positive or zero. The fact that the 2IFC/YN or 2IFC/SD ratios are less than the pre-

dictedV•- in detectiontasks suggeststhat subjectscannot make the necessary subtraction consistently.

In dis-

crimination tasks where the 2IFC/YN and 2IFC/SD ra-

tios are greater thanthe predictedVr•-, it is difficultto determine

the relative

contributions

of order

information

and memory variance to the ratios. When the SD/YN ratio is greater than 1, however, it is possible to sort

J. Acoust. Soc. Am., Vol. 57, No. 5, May 1975

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1167

Jesteadtand Sims: Decisionprocesses in frequencydiscrimination

out the two factors. In this case, the subject is assumed to be listening to both intervals in SD and 2IFC with the same memory effects in both paradigms. If the subject is making full use of the order information in 2IFC, the

2IFC/SD ratio shouldbe 2. 0. The mean 2IFC/SD ratio for both pure-tone conditions in Experiment i is 1.88. This suggests that the subjects are making better use of the order information in 2IFC in the pure-tone discrimination

task

than

in the

FM

detection

task.

C. The memory factor The data presented here suggest that there is greater memory variance in discrimination tasks than in detection tasks. The interpretation of this factor that seems most compatible with the other TSD assumptions is to consider assume

it as criterion that

the

variance.

criterion

It is reasonable

is more

stable

when

to

the two

signals produce sensations that differ qualitatively as well as quantitatively. A stable criterion based on a qualitative difference is, of course, a characteristic of

categorical perception (Studdert-Kennedy et al., Our interpretation

of the difference

and discrimination

tasks

has much

recent analysis of categorical

sounds(Miller et al.,

1970).

between detection in common

with

a

perception of nonspeech

1973). It shouldbe noted that

their analysis included detection of pure tones in noise as an example. VI.

CONCLUSIONS

The experiments

reported here indicate the following.

(1) The results obtained with pure-tone and FMsignals differ in many ways, even if the signals are presented in identical paradigms. The pure-tone and FM data differ in terms of overall performance, performance as a function of frequency, performance as a function of paradigm, and the nature of the psychometric function.

(2) The variability across signal conditions and paradigms is large with respect to the variability across subjects. The data for pure-tone discrimination in the

2IFC paradigm can be added to a growing list of highly consistent

2IFC

data collected

frequency-discrimination

in a number

of different

studies.

(3) The differences in frequency discrimination as a function of paradigm are in the direction predicted by

TSD, not in the direction predicted by Wickelgren (1969). (4) The effect of paradigm is larger than TSD would predict for both pure-tone and FM discrimination, but smaller than TSD would predict for FM detection. This suggests that memory variance is entering into the decision process in some paradigms in the discrimination task, but not in the detection task.

1167

ACKNOWLEDGMENTS

This research was supported by grants from the Na-

tional Institutes of Health, Public Health Service, U.S. Department of Health, Education and Welfare, and the National

Science

Foundation.

The authors

wish to ex-

press their appreciation to Robert C. Bilger, David M. Green, and Craig C. Wier for many helpful comments on earlier

versions of this paper.

*Now at the Department of Psychology and Social Relations, Harvard University, Cambridge, MA 02138.

lWe use rangeof modulation here to refer to the difference between the highest and lowest excursions of the carrier frequency. This usage is consistent with that of previous FM studies. The procedures for calibrating range of modulation were similar to those described by Feth, Wolf, and

Bilger (1969).

2Thefact thatwe usedan 8-Hz modulation rate, rather than the 2-Hz rate used by Shower and Biddulph (1931) and Filling (1958), may also accountfor the larger values of Af/f we obtained. The calibration procedure we followed, however, should have eliminated the potential modulation-rate effect (Feth, Wolf, and Bilger, 1969).

3Wehavenotfoundanydiscussionin the literature of the fact that this variability is inherent in ratios of estimates of performance. If the error associated with each performance estimate is normally distributed, then the error associated with the ratio estimate will have a Cauchy distribution (see Cramer, 1946, p. 246). This distribution has infinite variance and, as a result, the law of large numbers does not hold. Mean ratios across subjects or conditions should therefore show the same variability as individual ratios. In reality,

the use of small numbers of trials

tend to limit

the size of the errors

associated

and other factors with

individual

performance estimates, so the variance of ratio estimates is less than infinite and there is some small advantage in considering mean ratios. The inherent variability cannot be avoided because there is no way of comparing paradigms that does not implicitly or explicitly involve ratios.

4Thecalculationof the memoryvariancerelative to the sensation variance is as follows. Assume that both dsD and dyN are given by the difference in mean sensations divided by the square root of the sum of the variance terms.

In the

YN case, there are two variance terms, one representing the sensation variance in a single observation and the other representing memory variance. In the SD case, the memory variance is assumed to be negligible, but the observation variance is doubled. Therefore,

d,SD_ dyN ,

•/O+M ,/2 ß 0

where 0 is the variance of a single observation and M is the memory variance. It follows directly that

The ratio of dsD to dy• is equivalent to the ratio of the slopes of the psychometric functions.

(5) A similar pattern of paradigm effects is observed in the literature on pure-tone detection and intensity discrimination. This suggests that the additional source of variance is not related to a specific signal ensemble but is central in origin. The approach most compatible with TSD is to interpret the memory variance as criterion

variance

stable

and to assume

in detection

tasks

that

the criterion

than discrimination

is more

tasks.

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