Use of psychometric-function slopes for forward-masked tones to investigate cochlear nonlinearity

Kim S Schairer; Jessica Messersmith; Walt Jesteadt

doi:10.1121/1.2968686

. 2008 Oct;124(4):2196–2215. doi: 10.1121/1.2968686

Use of psychometric-function slopes for forward-masked tones to investigate cochlear nonlinearity¹

Kim S Schairer ^1,^b), Jessica Messersmith ¹, Walt Jesteadt ¹

PMCID: PMC2600619 NIHMSID: NIHMS59658 PMID: 19062859

Abstract

Schairer et al. [(2003). “Effects of peripheral nonlinearity on psychometric functions for forward-masked tones,” J. Acoust. Soc. Am. 133, 1560–1573] demonstrated that cochlear nonlinearity is reflected in psychometric-function (PF) slopes for 4 kHz forward-masked tones. The goals of the current study were to use PF slopes to compare the degree of compression between signal frequencies of 0.25 and 4 kHz in listeners with normal hearing (LNH), and between LNH and listeners with cochlear hearing loss (LHL). Forward-masked thresholds were estimated in LNH and LHL using on- and off-frequency maskers and 0.25 and 4 kHz signals in three experiments. PFs were reconstructed from adaptive-procedure data for each subject in each condition. Trends in PF slopes across conditions suggest comparable compression at 0.25 and 4 kHz, and potentially a wider bandwidth of compression in relative frequency at 0.25 kHz. This is consistent with other recent behavioral studies that revise earlier estimates of less compression at lower frequencies. The preliminary results in LHL demonstrate that PF slopes are abnormally steep at frequencies with HL, but are similar to those for LNH at frequencies with NH. Overall, the results are consistent with the notion that PF slopes reflect degree of cochlear nonlinearity and can be used as an additional measure of compression across frequency.

INTRODUCTION

The overall goal of the current set of experiments was to expand the results of Schairer et al. (2003b), which demonstrated that cochlear nonlinearity is reflected in slopes of psychometric functions (PFs) for forward-masked, 4 kHz tones. PF slopes are used here to investigate cochlear nonlinearity at 0.25 kHz in comparison to 4 kHz in listeners with normal hearing (LNH) and in listeners with cochlear hearing loss (LHL). Sections 1A, 1B, 1C, 1D describe how cochlear nonlinearity is reflected in forward masking, how cochlear nonlinearity is reflected in PF slopes, and how PF slopes can be used to investigate compression at low frequencies and in ears with HL.

Forward masking and cochlear nonlinearity

Forward masking refers to the condition in which the threshold for a short-duration signal is elevated in the presence of a preceding masker. Forward-masked thresholds have been used to estimate frequency selectivity (see, e.g., Nelson and Freyman, 1984) as well as auditory time constants or temporal resolution (see, e.g., Nelson and Pavlov, 1989; Nelson and Freyman, 1987; and Jesteadt et al., 1982). Many recent studies have used forward masking to assess the amount of cochlear compression or nonlinear basilar-membrane (BM) response growth (see, e.g., Lopez-Poveda et al., 2003; Nelson et al., 2001; Nelson and Schroder, 2004; Oxenham and Plack, 1997; Plack and Oxenham, 1998; Rosengard et al., 2005; Schairer et al., 2003b; and Williams and Bacon, 2005).

Forward masking is thought to reflect either temporal integration or adaptation, or some combination of both processes (Chatterjee, 1999; Oxenham, 2001; Plack and Oxenham, 1998). Temporal integration is conceptualized as an overlap of the internal representations of the signal and the masker that occurs centrally; it can also be thought of as “persistence” of neural activity after the masker offset. Adaptation is the reduction of activity or response to a signal after presentation of a masker. It may occur at different places in the auditory periphery, such as the synapse between the inner hair cells (IHCs) and the eighth nerve, and the involvement of neural adaptation is supported by a recent computer modeling study (Meddis and O’Mard, 2005). Peripheral adaptation cannot entirely account for the observed threshold shift, however, because forward masking can be obtained in individuals with cochlear implants in whom stimulation bypasses the cochlea and IHC-eighth nerve synapse (Chatterjee et al., 2006; Chatterjee, 1999; Shannon, 1990). Thus, forward masking is due not only to peripheral adaptation, but is almost certainly influenced by a retrocochlear process.

For sinusoidal forward maskers and signals, the amount of forward masking is greater when the masker and signal are at the same frequency (on frequency) than when the masker is at a different typically lower frequency (off frequency). Slopes of growth of masking (GOM), or the change in signal level at threshold for a given change in masker level, can be obtained in conditions in which the signal level is varied to estimate threshold in different fixed masker-level conditions [variable signal (VS)] or conditions in which the masker level is varied in different fixed signal level conditions [variable masker (VM)]. In VS conditions, slopes of GOM are less than 1 dB∕dB in on-frequency conditions and are similar at low and high frequencies in LNH (see, e.g., Jesteadt et al., 1982). Signal level at threshold increases at a faster rate for off-frequency than for on-frequency conditions at moderate masker levels (see, e.g., Luscher and Zwislocki, 1949 and Plack and Drga, 2003). Comparable effects can be observed by fixing the signal at a low level in a VM paradigm with different signal delays in different conditions using on- and off-frequency maskers (Nelson and Schroder, 2004; Nelson et al., 2001). The resulting temporal masking curves (TMCs) show the effects of both signal delay and masker frequency. GOM functions obtained with VS procedures are used in the current set of experiments.

BM response growth is nonlinear at characteristic frequency (CF) (see, e.g., Ruggero et al., 1997 and Yates et al., 1990). That is, if a tone with a frequency equal to CF is presented, the amount of BM deflection as a function of stimulus level is linear at low levels and gradually becomes compressive as stimulus level increases. If the recording is made from the same place on the BM, but a lower or higher frequency is presented, the response growth becomes more linear. Response growth also becomes more linear in ears with HL, presumably due to the loss of outer hair cell (OHC) function.

Although forward masking almost certainly has a retrocochlear contribution, it is thought that cochlear nonlinearity is reflected in slopes of GOM and TMC in on-frequency and off-frequency masker conditions. In a VS paradigm in LNH, the shallow GOM in the moderate masker-level range in on-frequency masker conditions is thought to be due to compression of the on-frequency masker (Plack and Oxenham, 1998). In a VM paradigm, the off-frequency masker condition has been used as a linear reference in a ratio of slopes of GOM for on- and off-frequency masker conditions to estimate the degree of compression in LNH and reduced compression in LHL (Oxenham and Plack, 1997). On-frequency TMCs are steeper and off-frequency TMCs are shallower in LNH, whereas the differences in TMC slopes decrease and functions become more parallel across masker frequencies in LHL (Nelson et al., 2001; Plack et al., 2004; Rosengard et al., 2005).

A three-stage model described by Plack and Oxenham (1998) can be used to predict the pattern of thresholds in a VS paradigm at various signal delays and masker levels. The model has a compressive nonlinearity as its first stage, followed by a sliding temporal integration window, and a decision process that compares the level at the output of the window to determine which interval contained the signal. The model uses a two-line approximation to the function representing compression, with linear growth of response at the signal frequency for low-level on-frequency maskers and constant compression above some breakpoint. The amount of compression can be determined by fitting a two-line function to the data.

There is a consensus in literature that the difference in on- and off-frequency GOM or TMC functions can be used to estimate compression at signal frequencies of 1 kHz or higher, although Wojtczak and Oxenham (2007) recently have questioned the necessary assumption that the rate of recovery is the same for on- and off-frequency forward maskers. The literature on compression at lower frequencies, reviewed here in a later section, suggests that the comparison of on- and off-frequency masking may be problematic because the assumption of linearity in the off-frequency reference condition may not be valid. Estimates of compression based on PF slopes for forward-masked tones do not require comparison of results obtained in on- and off-frequency conditions.

Cochlear nonlinearity reflected in PF slopes

PFs showing percent correct (PC) as a function of the level of the variable stimulus take different forms and have different meanings in VS and VM paradigms. In the VS paradigm, PC increases as the signal level increases. In a VM paradigm, PC decreases as the masker level increases, but the data can be fitted using the same equations and procedures. Schairer et al. (2003b) demonstrated that PFs obtained in a VS paradigm in LNH, with either on- or off-frequency forward maskers, have shallower slopes under conditions that result in large amounts of forward masking and interpreted the change in slope as a measure of compression. The rationale for this argument is shown in Fig. 1 (from Schairer et al., 2003b). In a VS paradigm, conditions that produce low signal levels at masked threshold will result in the signal passing through the more linear portion of the peripheral nonlinearity as the signal level varies during the threshold estimation procedure. PF slopes would be steep in this case because a small range of signal levels would be necessary to establish threshold. In conditions that produce higher signal levels at masked threshold, the signal will pass through the more compressive region of the function and will have to change by a greater amount to produce the same change at the output of the nonlinearity. In this case, the PF slope would be shallower. The nonlinearity is depicted in Fig. 1 as a two-line approximation, as represented in the Plack and Oxenham (1998) model. It is assumed that the true underlying nonlinearity has a more gradual transition into the compressive region (see, e.g., Neely and Jesteadt, 2005). The decrease in PF slope as a function of signal level at threshold should be the same for the on-frequency and off-frequency masker conditions in a VS paradigm, because for any given PF, signal level is the only varying parameter. The signal is “on frequency” by definition and will be affected by the nonlinearity at its place regardless of the masker frequency.

Relationship between compressive nonlinearity and psychometric-function (PF) slope (from Schairer et al., 2003b). In a forward-masking condition that produces a low signal level at threshold, the signal will pass through the more linear portion of the nonlinearity during threshold estimation. Changes in signal level at the input and output of the function will be similar, the standard deviation of the underlying distributions will be small, and the corresponding PF slope will be steep. In conditions that produce higher masked thresholds, the signal will pass through the more compressive region of the function. The signal level will have to change by a greater amount to produce the same change in signal level at the output of the nonlinearity, the underlying standard deviation will be larger, and the PF slopes will be shallower.

In a VM paradigm, however, PF slopes for on- and off-frequency conditions should be different because in this case, the varied parameter is the masker level, and the masker response growth will be different at the place of the signal depending on the relation between the masker and the signal frequency. In on-frequency conditions, results should be similar to the VS case because the masker will be compressed just as the signal is compressed. In off-frequency masker conditions, however, the masker response growth will be linear at the place of the signal and PF slopes should be steep and parallel, regardless of the masker level at the threshold. Schairer et al. (2003b) provided evidence to support these predictions in LNH using a 4 kHz signal and on- and off-frequency maskers.

Use of PF slopes to investigate compression at low frequencies

Behavioral studies have demonstrated a significant amount of compression at signal frequencies of 1 kHz and above (see, e.g., Nelson et al., 2001 and Oxenham and Plack, 1997), but earlier studies showed less compression at lower frequencies (see, e.g., Hicks and Bacon, 1999 and Plack and Oxenham, 2000). These studies relied on the assumption that response growth at the place of the signal is linear for off-frequency maskers. Several authors [Lopez-Poveda et al. (2003), Plack and Drga (2003), Plack et al. (2004), and Plack and O’Hanlon (2003)] have pointed out that if compression affects a wider range of frequencies relative to CF (Rhode and Cooper, 1996), then response growth of on-frequency and off-frequency maskers at the place of the signal would be similar and would confound methods that rely on the linearity of the off-frequency response. TMC and GOM methods that do not rely on the assumption of linear growth at the signal place for off-frequency maskers, or that use off-frequency growth at higher frequencies as a linear reference, have demonstrated low-frequency compression that is greater than in previous studies and is comparable to compression estimates for higher-frequency signals (see, e.g., Lopez-Poveda and Alves-Pinto, 2008; Lopez-Poveda et al., 2003; Plack and Drga, 2003; and Williams and Bacon, 2005). A recent report (Stainsby and Moore, 2006) suggests, however, that decay of forward masking is not independent of signal frequency and that it may not be appropriate to use a high-frequency reference to estimate compression at low frequencies. Additivity of masking (Plack and O’Hanlon, 2003; Plack et al., 2005) methods do not depend on an off-frequency masker condition or on uniform decay of forward masking across frequency and can be used to directly compare compression at low and high frequencies. In addition, Lopez-Poveda and Alves-Pinto (2008) described a method of using TMCs to estimate compression that does not require the assumption of uniform decay of forward masking across frequencies. In their new method, the comparison is made between TMC slopes obtained at two different probe levels within masker-frequency conditions. PF slopes are another measure of cochlear compression that requires no assumptions regarding the decay of forward masking or the degree of off-frequency compression and could provide an independent measure of the frequency range of compression at low frequencies.

Use of PF slopes to investigate compression in ears with hearing loss

There are few studies that address PFs for detection of tones in quiet in LHL. Marshall and Jesteadt (1986) obtained PFs for LHL and LNH for 0.5 and 4 kHz tones. PFs were estimated by straight-line least-squares fits weighted by number of trials for d^′ as a function of level. They reported no differences in PF slopes between the two groups. In contrast, Arehart et al. (1990) estimated PF slopes using linear regression and probit analysis for 0.5, 2, 4, and 8 kHz tones in quiet in groups of LHL and LNH. The slopes of the two groups overlapped, but the LHL had significantly steeper slopes in the 2 kHz condition and some LHL had abnormally high slopes across frequencies.

Thresholds for forward-masked tones under a number of different conditions have been reported for LHL due to OHC dysfunction. However, PFs for those conditions have not been reported. Following the logic described for LNH (Fig. 1), it is predicted that PF slopes for forward-masked tones in a VS condition should be steeper for LHL than for LNH. This is because LHL presumably lack the nonlinearity that is hypothesized to be responsible for the decrease in PF slope with masked threshold in the LNH. The function in Fig. 1 would be a straight line with a slope near 1.0, rather than a two-part function with a compressive region. Thus, any changes in the external signal across the range would be represented by the same amount of change internally, similar to conditions that would produce thresholds in the lower-level (linear) portion of the function for LNH. PFs should remain steep and parallel across the masker frequency and the threshold level.

EXPERIMENT 1: PSYCHOMETRIC FUNCTIONS FOR FORWARD-MASKED, LOW- AND HIGH-FREQUENCY TONES IN A VARIABLE-SIGNAL PARADIGM IN LISTENERS WITH NORMAL HEARING

The purpose of experiment 1 was to use PF slopes to test the hypothesis that compression is comparable at low- and high-signal frequencies. Note that the purpose is to compare relative compression between frequencies and not to estimate specific parameter or compression values. This method has the benefit that it does not rely on a comparison of off- and on-frequency conditions. Because both types of maskers should have a similar effect on the PF slope, based on the argument in the Introduction and in Schairer et al. (2003b), only on-frequency conditions were included. Forward masking was measured in on-frequency VS conditions at both 0.25 and 4 kHz in a group of LNH. It was predicted that PF slopes would decrease similarly as a function of signal threshold in both the 0.25 and 4 kHz conditions, suggesting comparable cochlear compression at these frequencies.

Subjects

Five paid adults, two males and three females, ages 19–32 years (mean=25.2, standard deviation [SD]=6.3) served as subjects. Three were college students, one was the first author, and one was a research associate from another laboratory. Hearing for the three college students had been screened at 0.5, 1, 2, and 4 kHz within the past year using the same two-interval forced-choice (2IFC) adaptive procedure used in the experiment. Thresholds were at or better than 15 dB sound pressure level (SPL) for all test frequencies, bilaterally, for each subject. Hearing for the other two subjects had been tested, using clinical procedures, as part of a research protocol for another laboratory. Hearing thresholds were at or better than 15 dB hearing level (dBHL), bilaterally, at the same frequencies for both subjects.

Stimuli and apparatus

Signals and maskers were all pure tones. Signal frequencies were 0.25 or 4 kHz, signal duration was 10 ms (5 ms rise∕fall), and signal delay was 10 ms (from offset of masker to onset of signal). A 10 ms delay was selected in order to provide a sufficient amount of masking (in comparison to decreased masking at longer delays) and to avoid abnormally steep PFs observed in some listeners in very short (e.g., 0 ms) signal delay conditions (as observed in Schairer et al., 2003b). The on-frequency forward masker duration was 200 ms (2 ms rise∕fall) and the maskers were presented at levels of 30, 50, 70, and 90 dB SPL in separate conditions. All stimuli were generated with ramps that were shaped using a half-cycle of a raised-cosine function. Thresholds for each signal in quiet also were obtained.

Stimuli were generated digitally at a sampling rate of 50 kHz using a Tucker-Davis Technologies (TDT) array processor (TDT AP2) and 16 bit digital-to-analog converters (TDT DD1). The forward masker was generated on one channel of the DD1, while the signal was generated on the other. The output of each channel was low-pass filtered at 20 kHz (TDT FT6) and attenuated (TDT PA4), then the outputs of the two channels were combined (TDT SM3) and presented monaurally to the left ear through a headphone buffer (TDT HB6), a remote passive attenuator in the sound treated room, and a Sennheiser HD 250 Linear II headphone. Parallel use of multiple attenuators, summers, and headphone buffers made it possible to simultaneously test up to four listeners. Subjects 078 and 175 were tested individually. Subjects 100, 102, and 108 were tested as a group.

Adaptive-procedure thresholds

Thresholds were obtained in a VS paradigm using a two-track 2IFC adaptive procedure with decision rules to estimate 71 PC (two-down, one-up) on one track and 87 PC (five-down, one-up) on the other track (Levitt, 1971), with a 4 dB step size. Two tracks with different decision rules were used to obtain a larger range of PCs for the PF fits. Threshold for each track was calculated as the mean of the reversal levels after the fourth reversal. Five 200-trial repetitions were obtained in each condition. The first repetition was excluded as practice. A total of 800 trials, or 400 trials per track, were available for further analysis. There were two exceptions. Four repetitions were obtained in the 0.25 kHz quiet-threshold condition for subject 102. After excluding the first repetition as practice, a total of 600 trials remained for this subject in this condition. For subject 175, one repetition in the 4 kHz, 90 dB masker condition yielded only one reversal after the fourth with which to calculate the threshold. This repetition was discarded and an extra repetition was obtained. Mean thresholds were calculated across repetitions for each track, and then the mean across the two tracks was calculated as the adaptive-procedure threshold for each subject in each condition. This threshold was an estimate of the level required for 79 PC.

Psychometric-function fits

All trials from both tracks were combined to fit PFs. The combined data were trimmed such that signal levels that were presented on <30 trials across both tracks and∕or associated with <50 PCs were excluded. As the signal level increased, subsequent signal levels were deleted after the first occurrence of 100 PC. The purpose was to remove multiple levels with associated PCs of 100 that would skew the fits to be shallower than they probably were, and to avoid nonmonotonic functions. PFs were fitted for each condition using Dai’s (1995) modification of the equation proposed by Egan et al. (1969), in which d^′=(I∕a)^b, where I is signal power, 10 log(a) is the signal level required for d^′=1, and b is the slope of a line in log d^′ versus signal level. Dai’s (1995) fitting procedure minimizes the deviation between expected and obtained proportion correct, expressed in units of χ². To provide a more familiar measure of goodness of fit in terms of variance accounted for, an r² was calculated for each PF. For a total of 50 PF fits, including the quiet-threshold conditions, the r² values ranged from 0.79 to 1.00 (mean=0.95). Note that the fitting procedure yields a measure of the threshold, 10 log(a), that is computed very differently than the adaptive threshold described above. The correlation between the two threshold types across frequency, masker level, and subject (not including quiet thresholds, but including all masked thresholds regardless of associated slopes) was 0.998.

Results and discussion

Adaptive-procedure thresholds

Jesteadt et al. (1982) noted that forward masking appears more uniform across frequency conditions when thresholds are plotted in units of sensation level (SL). Figure 2 shows mean masked threshold across subjects as a function of masker dB SPL in the left panel and mean amount of masking as a function of dB SL in the right panel. Amount of masking was calculated for each subject in each condition by subtracting the signal threshold in quiet from the masked thresholds. Thresholds for the masker in quiet were not obtained for this group and some members of the group were not available to return to run those conditions. The mean masker thresholds in quiet for the LNH in experiment 2 were therefore used to estimate masker levels in dB SL in experiment 1.Threshold increases as a function of masker level at a similar rate for both frequencies, but appears to be consistently greater in the 0.25 kHz condition when expressed in terms of masker dB SPL (left panel). However, when corrected by masker threshold in quiet, the amount of masking is similar across signal-frequency conditions (right panel) and is well described by a single line with a slope of 0.58 dB∕dB.

Mean masked adaptive threshold in dB SPL across listeners with normal hearing (LNH) as a function of masker dB SPL (left panel) and mean amount of masking (i.e., signal dB SL) as a function of masker dB SL (right panel) for 0.25 kHz (squares) and 4 kHz (triangles) signal conditions in experiment 1. The mean masker thresholds in quiet for the LNH in experiment 2 were used to estimate masker dB SL in experiment 1. The error bars represent +∕−1 standard deviation. The on-frequency forward masker duration was 200 ms, and maskers were presented at 30, 50, 70, and 90 dB SPL in separate conditions. Signal duration was 10 ms and signal delay was 10 ms, presented in a variable signal (VS) paradigm. The amount of masking increases as a function of masker level at a similar rate for both frequencies. The line in the right panel is fitted with all data points (across frequency) and has a slope of 0.58 dB∕dB.

Psychometric-function fits

Figure 3 shows fits to the individual data points for the 30 dB (low threshold) and 90 dB (high threshold) masker conditions for both signal frequencies for each subject. There does not appear to be a difference in the goodness of fit to the data points as a function of frequency or masker level. Thus, the trend of shallower slopes as a function of signal level cannot be accounted for by poorer PF fits at higher levels. Table 1 provides a summary of the PF parameters and r² values for each subject in each condition. Figure 4 shows PF slope as a function of PF threshold in dB SL for each subject and for the geometric mean across subjects (geometric mean slope as a function of arithmetic mean PF threshold). One data point from the 4 kHz, 30 dB masker condition is missing from the panel for subject 175 because the slope is excessive (5.6). Despite the variability across subjects, on average PF slope decreases as a function of threshold at a similar rate for both frequencies. The pattern of slope change with level is related to the form of the compressive nonlinearity. The function fitted to the mean data in Fig. 4 is the reciprocal of a quadratic compression function, as described by Neely and Jesteadt (2005).

Psychometric-function (PF) fits to the data points in the lowest (30 dB SPL) and highest (90 dB SPL) masker-level conditions for each subject in experiment 1. Data points are represented as in Fig. 2. The dashed and solid lines represent PFs in the 0.25 and 4 kHz signal conditions, respectively. PF threshold, slope, and goodness of fit (r²) parameters for all subjects and conditions (including those not shown here) are listed in Table 1.

Table 1.

Experiment 1 psychometric function (PF) parameters for listeners with normal hearing (LNH) in variable signal (VS), on-frequency forward-masking conditions [QT=quiet threshold; SD=standard deviation].

Signal frequency	Masker level	Parameter	078	100	102	108	175	Mean(SD)
0.25 kHz	QT	Threshold	30.84	27.64	38.37	32.36	31.39	32.12 (3.92)
		Slope	1.79	0.99	0.74	0.63	2.04	1.11
		r²	0.99	0.98	0.88	0.92	0.95
	30	Threshold	31.72	30.55	41.29	34.74	35.07	34.67 (4.17)
		Slope	0.80	1.31	1.34	0.69	1.13	1.02
		r²	0.99	0.99	0.87	0.99	0.96
	50	Threshold	43.14	45.33	48.51	44.34	45.09	45.28 (2.0)
		Slope	0.72	0.57	1.10	0.37	1.25	0.73
		r²	0.98	0.98	0.97	0.94	1.0
	70	Threshold	53.20	56.42	59.29	57.74	55.14	56.36 (2.34)
		Slope	0.48	0.47	0.50	0.28	0.73	0.47
		r²	0.96	0.89	0.93	0.82	0.92
	90	Threshold	64.40	68.00	75.77	72.74	66.24	69.43 (4.71)
		Slope	0.37	0.35	0.38	0.27	0.76	0.40
		r²	0.93	0.96	0.98	0.89	0.98
4 kHz	QT	Threshold	17.15	13.28	22.01	16.91	15.27	16.93 (3.24)
		Slope	1.05	1.93	1.16	2.42	1.36	1.58
		r²	0.98	1.00	0.98	0.99	1.00
	30	Threshold	27.68	23.71	30.10	27.67	27.57	27.35 (2.30)
		Slope	1.24	2.23	0.83	0.52	5.57	1.46
		r²	0.99	0.99	0.85	0.84	1.00
	50	Threshold	36.16	30.66	43.39	37.32	33.04	36.11 (4.84)
		Slope	1.11	0.73	0.78	0.30	0.91	0.71
		r²	0.99	0.92	0.97	0.86	0.95
	70	Threshold	40.94	38.86	51.49	50.56	39.08	44.18
		Slope	1.07	0.52	0.53	0.20	0.83	0.55
		r²	0.99	0.96	0.93	0.79	0.95
	90	Threshold	52.09	50.42	73.14	77.92	53.04	61.32 (13.11)
		Slope	1.00	0.39	0.40	0.28	0.30	0.42
		r²	0.99	0.98	0.90	0.95	0.95

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Psychometric-function (PF) slope as a function of PF threshold in dB SL for each subject and the geometric mean across subjects for experiment 1. Frequencies are represented as in Fig. 2. On average, although there is variability across subjects, PF slope decreases as a function of masked threshold for both frequencies at approximately the same rate. The function fitted to the geometric mean data is the reciprocal of the quadratic compression function described by Neely and Jesteadt (2005).

In summary, thresholds in on-frequency VS conditions increased as a function of masker level and PF slopes decreased as a function of signal threshold similarly for both the 0.25 and 4 kHz signals. Results suggest comparable cochlear on-frequency compression at these two frequencies.

EXPERIMENT 2: PSYCHOMETRIC FUNCTIONS FOR FORWARD-MASKED, LOW- AND HIGH-FREQUENCY TONES IN A VARIABLE-MASKER PARADIGM IN LISTENERS WITH NORMAL HEARING

The purpose of experiment 2 was to test the hypothesis that the bandwidth of compression at 0.25 Hz is wider than at 4 kHz. VS conditions do not provide information about the range of frequencies that are compressed at each CF because on- and off-frequency conditions produce similar PF slopes. However, for any given PF in a VM condition, the varied parameter is the masker level, which will grow compressively at the place of the signal in the on-frequency condition, and linearly or less compressively at the place of the signal for the off-frequency condition. If the bandwidth of compression at 0.25 kHz is wider than at 4 kHz signal, then PF slopes might decrease as a function of threshold in both on-frequency and off-frequency conditions, because the off-frequency masker may grow compressively at the place of the signal, just as the on-frequency masker does. Thus, the prediction is that in the 4 kHz signal condition, PF slopes will decrease as a function of masker threshold for the on-frequency conditions and will remain steep for off-frequency conditions; in the 0.25 kHz signal condition, PF slopes will decrease as a function of threshold similarly in on- and off-frequency conditions.