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J Am Acad Audiol 10 : 248-260 (1999) Loudness Scaling Revisited Claus Elberling* Abstract The present work was undertaken in an attempt to evaluate whether it is reasonable to expect that categorical loudness scaling can provide useful information for nonlinear hearing aid fitting . Normative data from seven scaling procedures show that the individual procedures relate the perceptual categories differently to sound level and with a substantial betweensubject variance . Hearing-impaired data from four studies demonstrate that the inverse slope of the loudness function varies linearly with hearing loss and with a constant variance . In relation to hearing aid fitting, the slope can, in most cases, be predicted from the hearing loss with an accuracy within the range of a normal finetuning . For the fitting of nonlinear hearing aids, the statistical properties of both normal and impaired loudness functions are equally important. The present analysis strongly suggests that categorical loudness scaling cannot, in general, provide significant information for the fitting process. Key Words: Categorical loudness scaling, hearing aid fitting, loudness restoration, loudness scaling procedures, normative reference, slope of loudness function Abbreviations : a = slope of fitted straight line, CB = critical band, COVXY = covariance between variables x and y, CR = compression ratio, GL = hearing aid insertion gain at input level L, HTL = hearing threshold level, I/O = input/output, LGOB = loudness growth in 1/2-octave bands, RETSPL = reference threshold sound pressure level, S = slope of loudness growth function, VARX = variance of variable x, WDRC = wide dynamic range compression. everal dimensions of a sensorineural hearing loss are important for hearing aid fitting (e .g ., sensitivity, dynamic range [recruitment], and frequency resolution) . The recent focus on dynamic range has resulted in a seemingly "logical" signal processing strategy called amplitude compression. Various compression schemes have been proposed for the restoration of loudness, and it has been suggested that a (preferably multichannel) hearing aid with flexible compressors adjusted according to the measured impaired loudness can alleviate major problems associated with hearing impairment . However, for a variety of reasons (e .g., loudness summation and the actual performance of the hearing aid), most straightforward "recruitment compensation methods" do not restore normal loudness in realistic sound environments . Further, the hearing-impaired person may not necessarily prefer such an ampli- S *OTICON Research Centre, Eriksholm, Snekkersten, Denmark Reprint requests : Claus Elberling, OTICON Research Centre, Eriksholm, 243 Kongevejen, DK 3070 Snekkersten, Denmark 248 fication strategy over another, because, in sensorineural hearing loss, auditory dimensions other than loudness are affected . This paper will not discuss whether a hearing aid fitted to perform loudness restoration performs as intended with speech signals or in real acoustic environments, nor will it discuss whether loudness restoration in itself is a useful way to alleviate hearing impairment . These issues are important and should be properly addressed through adequate experiments, including field testing. Instead, the present work will focus on fundamental issues in loudness scaling and some of the underlying assumptions that are made in relation to hearing aid fitting in general. Over the years, much effort has been made to develop adequate methods to measure loudness, and a series of different loudness scaling procedures have been proposed especially for hearing aid fitting, rather than for diagnostic purposes . There is no doubt that the loudness dimension is important for the design of hearing aids and their signal processing algorithms, as well as for the individual fitting. At the very least, loudness addresses issues such as audibility at low sound levels and uncomfortable exposure to high sound levels . Loudness Scaling Revisited/Elberling Fitting procedures that are based on individual loudness scaling measurements have been recommended, especially for the manage- ment of nonlinear hearing aids (e .g., Independent Hearing Aid Fitting Forum [IHAFF ; Valente and Van Vliet, 1997], ScalAdapt [Kiessling et al, 1995], and Real Ear Loudness Mapping [RELM ; Humes et al, 1994] ) . There may be at least three advantages of applying loudness scaling in the fitting process : (1) it will focus on audibility and listening comfort, both at low and at high sound levels (re : the above) ; (2) it will make the hearing-impaired person feel more personally involved ; and (3) it may lead to a more accurate initial setting of the hearing aid . Most of the claims that have been made about loudness scaling and hearing aid fitting have more or less ignored the first two arguments . However, if the use of individual loudness scaling turns out to be advantageous for the fitting of hearing aids, we must be aware that there might be several reasons for such a positive effect . The present work is an attempt to address the latter argument, namely, the initial setting of the hearing aid . This is done not as an experimental clinical evaluation but as an analysis of the underlying structure of loudness scaling data . Analyses addressing the accuracy of such data have recently been presented by Beattie et al (1997), Palmer and Lindley (1998), and Rasmussen et al (1998) . Published loudness scaling data from subjects with normal hearing and with hearing impairment indicate both differences and similarities among data obtained with the different scaling procedures . Therefore, this paper will analyze loudness scaling data published in the literature and ask the question whether such data are adequate for getting a more accurate or acceptable initial hearing aid setting . The analysis presented herein is not based on separate experiments designed for the present purpose but uses loudness scaling data published within the last decade by different groups of researchers . The analysis is in three parts : (1) normative data, (2) hearing-impaired data, and (3) synthesis . ANALYSIS Normative Data Variance across Different Scaling Procedures For each loudness scaling procedure, a set of data obtained from a well-defined group of subjects with normal hearing is called the norma- tive reference . Most published scaling procedures show in graphic format the corresponding normative reference that subsequently is used when evaluating individual data obtained with that procedure. Due to inherent differences between the procedures, the normative references cannot easily be compared . The differences may be related to one or more of the following: stimulus parameters, randomization of stimulus levels and frequencies, anchoring and ceiling effects, application of loudness categories, instructions to the test subjects, and more . However, newcomers would intuitively expect that for comparable parameters, the procedures would result in about the same loudness rating . In order to investigate whether this holds true, the normative references from seven different procedures are compared in the following section. All of the procedures make use of narrowband stimuli and have some commonality regarding use of loudness categories (i .e ., they make use of all or a superset of the categories : too soft, soft, comfortable [intermediate or okay], loud, very loud, and too loud). Some of the procedures give stimulus level directly in dB HL and others in dB SPL measured for a specific transducer in a specific coupler. For the present comparison, all data sets are referenced to the dB HL scale. The transformation from dB SPL to dB HL of the individual data set is based on the information available for each procedure. Below, the seven procedures are briefly summarized and the relevant specifications given: 1 . Allen et al (1990) used the loudness growth in 1/2-octave bands (LGOB) procedure to rate the loudness in seven categories (not heard, very soft, soft, okay [comfortable], loud, very loud, and too loud) from 11 subjects with normal hearing with a random level and frequency presentation of 250, 500, 1000, 2000, and 4000 Hz, 1/-octave wide noise stimuli. Stimulus levels were given in dB SPL measured in the Bruel & Kjaer 4157 coupler (IEC 711) . First, the group average data were read from Figure 4 in Allen et al (1990) and since the "not heard" and "too loud" data were not plotted in the figure, they were estimated from the shapes of the given input/output (I/0)-functions in their Figure 3 and the transition response levels in their Figure 4. Second, because 1/2-octave is different from the width of the critical band (CB), bandwidth corrections (i .e ., 10*Log[1/2octave/CB]) were calculated for each frequency. Third, to transform the dB SPL 249 Journal of the American Academy of Audiology/Volume 10, Number 5, May 1999 values to dB HL, the reference threshold levels (RETSPLs) (ISO, 1994) for insert earphones in the IEC 711 coupler were applied. Finally, since the values at 250 Hz referenced to dB HL deviate significantly from those at the other frequencies, the 250-Hz data were excluded in the final analysis . The data at 500, 1000, 2000, and 4000 Hz were averaged over frequency yielding the final normative reference for the LGOB procedure. 2 . Elberling and Nielsen (1993) used the CAR procedure to rate the loudness in seven categories (not heard, very soft, soft, okay [comfortable], loud, very loud, and too loud) from 10 subjects with normal hearing with a random level presentation of 500- and 2000-Hz pure-tone stimuli. On a dB HL scale, there were no significant differences between the results from the two frequencies; therefore, the data were pooled to give the normative reference for the CAR procedure. 3. Kiessling et al (1993) used the direct loudness scaling procedure to rate the loudness in 13 categories (seven labeled: not heard, very soft, soft, middle loud [comfortable], loud, very loud, and uncomfortably loud and six nonlabeled, interleaved) on a 50-point scale from 10 subjects with normal hearing with a quasi-random level presentation of 1/3-octave filtered noise stimuli at the center frequencies 500, 1000, 2000, and 4000 Hz . Stimulus level was given in dB HL and no significant difference over frequency was observed . The average data (pooled over frequency) were read from Figure 9a in Kiessling et al (1993) and used to represent the normative reference for this procedure. 4. Hohmann and Kollmeier (1995) used the Horfeldskalierung procedure to rate the loudness in five categories (very soft, soft, intermediate [comfortable], loud, and very loud) and 10 response possibilities from 26 subjects with normal hearing with a quasirandom level presentation of 1/3-octave filtered noise stimuli at the center frequencies 250, 500, 1000, 2000, and 4000 Hz . Each measured data set (i.e ., category vs stimulus level [in dB HL]) was fitted to a straight line giving the slope, m, and the level corresponding to the comfortable (intermediate) category, L25,-Table 2 in Hohmann and Kollmeier (1995) . For the present analysis, the average over frequency of the two parameters was calculated and subsequently used to compute the stimulus level for each category. This result is used to describe the normative reference for this procedure. 5 . Launer (1995) used a categorical scaling procedure to rate the loudness in seven categories (inaudible [not heard], very soft, soft, intermediate [comfortable], loud, very loud, and too loud) and 11 response possibilities from nine subjects with normal hearing with a quasi-random level presentation of one critical band "frozen" noise stimuli centered at 1370, 1600, 1850, 2150, 2500, and 2925 Hz . Each measured data set (i.e ., category vs stimulus level [in dB HL]) was fitted to a straight line giving the slope, m, and the level corresponding to the comfortable (intermediate) category, L25,-Table 5.2 in Launer (1995) . For the present analysis, the average over frequency of the two parameters was calculated and subsequently used to compute the stimulus level for each category. This result is used to represent the normative reference of this categorical scaling procedure . 6. Ricketts and Bentler (1996) used a categorical scaling procedure to rate the loudness in nine categories (cannot hear [not heard], very soft, soft, comfortable but slightly soft, comfortable, comfortable but slightly loud, loud, very loud, and too loud) from 20 subjects with normal hearing with a random level presentation of 1/3-octave noise stimuli centered at 500 and 3150 Hz . The average data of the response categories from soft to loud were presented as bar graphs in Figures la and lb in Ricketts and Bentler (1996) . First, for each frequency, the average values in dB SPL were read from the two graphs . Second, to approximately convert the dB SPL values measured with insert earphones on a Zwislocki coupler to dB HL, the RETSPLs (ISO, 1994) for insert earphones in the ear simulator (IEC, 1981) were applied. Finally, the dB HL values for 500 and 3150 Hz were averaged into the normative reference for this categorical scaling procedure . 7. Cox et al (1997) used the Contour Test of Loudness Perception to rate the loudness in seven categories (very soft, soft, comfortable but slightly soft, comfortable, comfortable but slightly loud, loud but okay, and uncomfortably loud) from 45 subjects with normal hearing with an ascending level presentation of warble tone stimuli at the center frequencies 250, 500, 1000, 2000, 3000, Loudness Scaling Revisited/Elberling and 4000 Hz . For each frequency and each loudness category, the mean level (dB SPL) was calculated and presented in Table 2 in Cox et al (1997) . To transform the dB SPL values to dB HL, the RETSPLs (ISO, 1994) for insert earphones in the HA-1 coupler (IEC, 1973) were applied . The values over frequency were averaged and used as the normative reference for the Contour Test . The normative reference data corresponding to the common categories very soft, soft, comfortable, loud, very loud, and too loud from these seven procedures are presented in dB HL in Table 1 and are plotted in Figure 1 . From these data, we can conclude that the stimulus level for the individual category varies significantly over the different procedures . For instance, the sound level corresponding to comfortable, which is supposed to be located at a medium loudness level, varies from about 63 to 89 dB (i .e ., a range of 26 dB for the procedures used in the present comparison). Discussion . The exact acoustic calibration that underlies all of the seven referred sets of data influences the above comparison and conclusions . For three of the referred sets of data (1, 6, and 7), it was further necessary to transform the original sound levels from dB SPL to dB HL based on the available information provided in each of the original papers . It is difficult to evaluate precisely the effect of this inherent uncertainty, but it is not likely that calibration and/or transformation errors would exceed 5 dB in the frequency range 250 to 4000 Hz . Therefore, the conclusion above seems warranted . If one used loudness scaling data to set the gain of a hearing aid at, for example, the comfortable loudness level, then the choice of scaling procedure would have a significant impact on the sound level at which this gain is set and, Table 1 Very loud Too loud VERYLOUD LOUD COMFORT o Allen et al, 1990 SOFT VERYSOFT NOTHEARD 0 0 20 . 40 , 60 , " Eberling and Nielsen, 1993 c Kiessting et al, 1993 0 Hohmann and Kollmeier, 1995 it Launer, 1995 V Ricketts and Gentler, 1996 + Cox et al, 1997 R0 100 SOUND LEVEL (dB HL) 120 Figure 1 The normative reference (i .e ., categories versus sound level [dB HLI) for seven loudness scaling procedures . therefore, on the electroacoustic characteristics of the fitted aid . The large differences between the normative references are most likely caused by differences in the psychoacoustic methods applied as recently reviewed by Jenstad et al (1997), Beattie et al (1997), and Ricketts (1997) . Jenstad et al investigated the effect of presentation mode and the influence of a preceding reference stimulus of maximal level. Beattie et al discussed presentation mode and the short-term reliability of the IHAFF contour test (Valente and Van Vliet, 1997). Ricketts discussed factors such as number and spacing of categories, type and bandwidth of the stimulus, presentation parameters, and instruction . However, as mentioned above, it is not the aim of this presentation to discuss more or less obvious procedural differences but to demonstrate the resulting effects and thereby to increase the readers' awareness about the existing differences between published normative references and some of the practical implications . Elberling and Nielsen (1993) Kiessling et al (1993) Hohmann and Kollmeier (1995) 34 .4 58 .9 78 .6 90 .2 39 .4 69 .3 89 .1 103 .8 22 .5 48 .8 64 .0 75 .0 42 .9 102 .6 125 .3 92 .5 106 .3 97 .6 6 Normative References for Seven Loudness Scaling Procedures Allen et al (1990) Very soft Soft Comfortable Loud TeoLo- 115.6 87 .5 57 .0 71 .1 85 .2 99 .3 Launer (1995) Ricketts and Bentler (1996) Cox et al (1997) 29 .4 20 .5 20 .3 66 .4 84 .9 62 .9 81 .8 67 .0 91 .9 47 .9 103 .4 112 .7 35 .5 - 39 .3 - 101 .0 The table shows categories versus sound level in dB HL . 251 Journal of the American Academy of Audiology/Volume 10, Number 5, May 1999 Variance across Normal-hearing Subjects In relation to hearing aid fitting, the loudness scaling data obtained from a hearingimpaired person (with sensorineural hearing loss) are almost always evaluated relative to the normative reference, either directly by comparing unaided and aided loudness data to the normative reference or indirectly by setting up a fitting target based on normative relations, as, for instance, loudness growth and speech levels . Thus, in its clinical use, loudness scaling does not differ from other audiologic measures, as, for instance, the pure-tone threshold. Here, 0 dB HL constitutes the normative reference (standardized in the ANSI S3 .6 [19891 and ISO 389 [19911 standards) . However, whereas many other normative references are based on data that demonstrate a limited variance over the group of normal-hearing subjects, the clinical loudness scaling methods described in the literature all show a considerable variance . To illustrate this, data from three different category rating methods are compared . The intersubject variance can be evaluated in many ways as it affects different parameters of the resulting loudness growth function . However, to restrict the analysis and focus attention on what may be clinically most important, the variance at a medium loudness level (i .e ., comfortable) will be presented: 1. Elberling and Nielsen (1993) measured the loudness functions for two frequencies on 10 subjects with normal hearing with the CAR procedure. All data are shown in Figure 2. TOOLOUD VERYLOUD LOUD 100 1~ I 80- VERYSOFT NOTNEARD 60- Table 2 Between-subject Variance of the Sound Level in dB HL Corresponding to 40 "Comfortable" in Subjects with Normal Hearing for Three Loudness Scaling Procedures 3'0' 1 20 100 Figure 2 20 ~T~T~T 40 60 80 100 SOUND LEVEL (dB HL) All data points obtained from loudness scal- Elberling and Hohmann and Kollmeier (1995) et al (1997) 9.0 36 8.0 32 10 .5 42 Nielsen (1993) 120 ing at two frequencies in 10 subjects with normal hearing (Elberling and Nielsen, 1993). The fitted exponential functions for the data points for one frequency in two subjects are indicated by (MT) and (HTW). 252 For the three procedures, the calculated standard deviations are given in Table 2, together with the normative 95 percent intervals . (Under the assumption of a Gaussian distribution of the data, the 95 percent interval corresponds to ±2 SD .) For the three procedures, this interval is considerable and varies from 32 to 42 dB . 70- COMF SOFT The rated categories versus stimulus level were fitted to an exponential function by the method of least squares and subsequently the stimulus levels were calculated corresponding to "okay" (comfortable) on each individual curve. Over the 20 observations (10 subjects and two frequencies), a mean of 87 .4 dB HL and a standard deviation of 9.0 dB was found. 2. Hohmann and Kollmeier (1995) rated the loudness from 26 subjects with normal hearing with the Horfeldskalierung procedure. Each measured data set (i .e ., category vs stimulus level for each test frequency) was fitted to a straight line giving the slope and the level corresponding to the intermediate (comfortable) category, L25 (Hohmann and Kollmeier, 1995, Table 2) . Over the five frequencies (130 observations), we calculated a mean of 71 .1 dB HL and a standard deviation of 8.0 dB . 3. Cox et al (1997) rated the loudness at six frequencies on 45 subjects with normal hearing with the Contour Test . For each frequency and each loudness category, the mean level and standard deviation were calculated . Over the six frequencies (270 observations) and at comfortable loudness level, we calculated (in dB HL) a mean of 67 .0 dB HL and a standard deviation of 10 .5 dB . SD (dB) 95% interval (dB) Cox The table shows both the standard deviations and the 95 intervals (±2 SD). Loudness Scaling Revisited/Elberling Discussion. The loudness scaling procedures used in this analysis are believed to constitute a representative sample of the clinical methods presented within the last decade or so . It is concluded that the normative reference is not just a fixed number (at any specific loudness level) but covers a substantial interindividual range . A significant uncertainty exists, which at the comfortable level corresponds to a range of about 35 dB . As described later, the normative reference is the basis for calculating target gain and/or compression ratio (CR) when individual loudness scaling data are used for fitting hearing aids . Therefore, precision of the target gain is influenced not only by uncertainty of the loudness scaling data acquired from the individual hearing-impaired subject but also, and to the same extent, by uncertainty in the normative reference. It is for this reason that the evaluation presented herein is important for the application of individual loudness scaling data in hearing aid fitting . Figure 3 Normalized slope versus hearing loss obtained from 10 subjects with normal hearing and 29 subjects with impaired hearing tested at two frequencies (from Elberling and Nielsen, 1993). The thick line is the exponential fit to the data and the thin lines indicate the spread of the data . Hearing-impaired Data Variance across Different Scaling Procedures Several researchers have stated that the loudness function that corresponds to a sensorineural hearing loss cannot be estimated from the hearing threshold but has to be measured (e .g ., Launer et al [19961) . This statement is based on the observation that the loudness scaling parameters (e .g ., the slope [S] of measured loudness growth functions) display an increased variance with increasing hearing loss and may be only weakly correlated with the hearing threshold (e .g., Hohmann and Kollmeier [1995]) . An example of the increasing variance with hearing loss is given in Figure 3 . In the figure, normalized slope (i .e ., the slope of the individual loudness growth function) relative to the normative reference is expressed . The advent of nonlinear hearing aids with wide dynamic range compression (WDRC) has really spurred a "bandwagon" interest in the use of loudness scaling for hearing aid fitting. However, it would be relevant here to mention the pioneer work by Pascoe (1978), which early on demonstrated the important link between the fitting of linear hearing aids with output limiting and the residual dynamic range of the hearing impaired . In the present context, the difference between a linear and a nonlinear hearing aid is the 1/O function, which, for loudness restoration in a nonlinear hearing aid, should mirror the normalized slope, S1 . In such a nonlinear hearing aid, the interesting part of the I/O function, which performs linear compression, is often charac- 120 100 80 60 40 20 0 Figure 4 A loudness model for normal and impaired hearing. The diagonal line indicates normal hearing, whereas the three other lines indicate loudness growth functions with different slopes (S) but all corresponding to a 70 dB HL hearing loss (HTL .) . For loudness restoration, the target gain is proportional to 1/S, indicated at a 60 dB HL sound level. 253 Journal of the American Academy of Audiology/Volume 10, Number 5, May 1999 0.0 1 0 20 40 60 Hearing Loss (dB HL) 80 0.0 ; 100 0 D 1.6 20 40 r 60 Hearing Loss (dB HL) 80 100 1.4 1.2 0.4 0.2 0.04 0 20 0.0 40 60 Hearing Loss (dB HL) Hearing Loss (dB HL) Figure 5A-D Normalized inverse slope as a function of hearing loss from four different procedures . A straight line is fitted to each data set and also ±1 SD lines are shown . A, Elberling and Nielsen (1993) ; B, Kiessling (1995) ; C, Launer et al (1996) ; and D, Ricketts and Bentler (1996) . terized by the CR equal to the normalized slope, S;,-thus, CR = S;. Another way is to characterize the I/O function by a set of gain values at two input levels, for example, a low and a high level. Thus, if the gain at a low input level (50 dB SPL) is G50 and the gain at a high input level (80 dB SPL) is Ggo, then the relationship between G5o, G8o, and the CR (and S.) is : 1/CR = 1/ S; = 1 + (G80 - G5o)/30 (1) This formula demonstrates that gain is inversely proportional to both CR and S;. Another way to demonstrate this relationship is shown graphically in Figure 4. Here, normal loudness is indicated by the diagonal line-the reference-and the impaired loudness by the sloping line starting at the x-axis at 70 dB HL (hearing 254 threshold level, HTL;) . A target gain is indicated at 60 dB HL, close to the average medium loudness level for normal-hearing subjects . Now, the steeper the impaired loudness growth function the lower the target gain, and the shallower the loudness growth function the higher the gain (i .e ., an inverse relationship between slope and gain). The variance in slope of the loudness functions and the practical consequences for hearing aid fitting is probably more easily understood in relation to gain than to CR by most hearing aid researchers. Further, the output level of the hearing aid is dependent on the input level and the applied hearing aid gain. It is for these reasons that the hearing aid gain is more relevant than CR and therefore is used in the following explanation: Loudness Scaling Revisited/Elberling For a fixed hearing loss, the observed variance of 1/S will be translated directly to variance in gain . The data from Figure 3 are, therefore, first normalized relative to normal hearing (Sno mal hearing = 1) and thereafter plotted with the inverse slope, 1/S,, as a function of hearing loss in Figure 5A . A straight line is fitted to the data by the method of least squares . The line intersects the y-axis at y = 1, the normalized, average value of inverse slope, 1/Sn, for subjects with normal hearing . As a first order of approximation, it appears that the data can be described with a linear relationship between 1/S and HTL and with a constant variance of 1/S across hearing loss . It should be noted that the variance of 1/S now appears to be about the same for both subjects with normal and impaired hearing! col-a fact that was considered by the authors not to be of importance . Variance of Inverse Slope across Hearing-impaired Subjects The correlation coefficients in Table 3 indicate that only about 55 percent of the variance of 1/S in the data (mean of r2) can be explained by the straight line approximating the relationship between 1/S. and HTL. Therefore, it is obvious that a significant contribution to this variance comes from other sources than the pure-tone hearing loss . If we accept the linear model as a reasonable descriptor of the acquired data, we have achieved a common framework for the evaluation of the practical consequences of the observed variance . Therefore, the four different data sets are submitted to the following comparison, where the HTL variable is grouped into the categories 10, 30, 50, 70, and 90 dB HL . In Figure 6, the data sets are plotted with this grouping . The graph demonstrates remarkable similarities across the different data sets, although differences also are noted, especially related to the difference in a (see Table 3) . However, in order to make a unified presentation, a weighted average of the different data sets is calculated (by Bayesian inference ; Box and Tiao, 1973), and the (weighted) mean and ± 1 standard deviation are plotted in Figure 7. This figure is based on more than 1000 individual observations and demonstrates that the variance of 1/S is approximately constant, independent of hearing loss . It also demonstrates that the linear description is not valid for severe to profound hearing losses . This latter observation indicates a change in rate of reduction in dynamic range when the hearing loss exceeds about 70 dB HL . Since the variance of 1/S is approximately constant over hearing loss, the data from the individual data set can be taken relative to its regression line and, thereafter, all data can be Similarly, the data from subjects with normal and impaired hearing in the studies by Kiessling (1995), Launer et al (1996), and Ricketts and Bentler (1996) have been analyzed and the results are plotted in Figure 5B-D . The key parameters for the fitted data are presented in Table 3 (i .e ., the number of data points, N, the correlation coefficient, r, the slope of the fitted line, a, and the estimated standard deviation, SD). Discussion . The data from the four different procedures (Fig. 5A-D) appear reasonably similar and as a first-order approximation seem to follow the suggested linear relationship between inverse slope and hearing loss . The data sets display roughly the same variance of 1/S about the regression lines as a function of hearing loss, also including normal hearing. The exception is the data from Ricketts and Bentler (1996) in Figure 5D, where the variance of 1/S is significantly lower for subjects both with normal hearing and hearing impairment . The reason for a lower variance is not clear and the specifics of the applied test procedure do not explain this finding. However, it appears that the test subjects had previous experience with the test proto- Table 3 Key Parameters of the Fitted Lines in Figure 5A-D Elberling and Nielsen (1993) Number, N 78 Slope of line, a Estimated SD -0 .0091 0 .18 Correlation, r 0.78 Kiessling (1995) 550 0.67 -0 .0072 0 .19 Launer et al (1996) 284 0.65 -0 .0079 0 .20 Ricketts and Bentler (1996) 94 0.85 -0 .0084 0 .13 Journal of the American Academy of Audiology/Volume 10, Number 5, May 1999 1 .6, 1 .4- 1 .2E e z 1 .0Launeretal (N = 284) Rickett and Bentler (N = 93) CL 0.8- W 0 E2 0.6 C 0.4Elberling and Nielsen (N = 78) 0.20.0 20 0 40 60 Hearing Loss (dB HL) 80 Figure 6 Comparison ofthe four different data sets from Figure 5A-D. In this graph, the data are grouped into five hearing loss classes and the mean value for each data set is plotted for each class. collapsed over hearing loss . The observed distribution of deviations of inverse slope, 1/S;, from expectations (the individual regression lines) can thus be produced as shown in Figure 8. If we go back to formula 1 and vary the gain for high input, G80, for instance, ±5 dBas would be relevant during finetuning of a hearing aid-this variation in 1/S; would correspond to ±0 .17 . Now, with reference to Figure 8, a range of deviations of ±0 .17 covers about 70 percent of the cases, with about 12 percent of the 0.6 h 0.2 0.0 0 20 40 60 Hearing Loss (dB HL) 80 100 Figure 7 Weighted means and ±1 SD of the data underlying Figure 6. Two straight lines (dotted) are fitted to the mean values : one for the values at 10, 30, 50, and 70 dB HL and another for the values at 70 and 90 dB HL . 256 -1.00 -0.80 -0.60 -r~ -0.40 -0.20 0.00 0.20 y rm'as°' ., - 0.40 0.60 0.80 Deviations of Inverse Slope (I/S) 1 .00 1.20 Figure 8 Observed distribution o£ deviation of the inverse slope from regression line . The deviations are collapsed over hearing loss and over the four procedures in Figure 5. 100 0.4 0 i--n-- observed values being below this range and about 17 percent being above it . Lower values mean steeper loudness growth and higher values mean shallower loudness growth . About 70 percent of the observations would, therefore, be covered by a finetuning of -} 5 dB, and, in these cases, the target CR could be obtained without individual loudness scaling testing. Discussion. It is obvious that the observed distribution in Figure 8 is skewed . Some of the asymmetry may be attributed to a ceiling effect for lower values (toward -1 .0 in Fig. 8) . For higher values, the effect of conductive components in the hearing loss or contribution from presbycusis (Knight and Margolis, 1984) is known to give shallower loudness growth function than in pure sensorineural hearing losses (toward higher values in Fig. 8) . Thus, the observed skewness could be a result of an inhomogeneous diagnostic classification . However, there are also other possible causes-for instance, a different etiology underlying the hearing losses and a different pattern of damage in the Organ of Corti, as discussed by Launer et al (1996) . Two points should be stressed : First, as indicated in Figure 7, the variance of 1/S is constant across hearing loss, but only as a first-order approximation . The Kruskal-Wallis one-way analysis of variance test (Siegel, 1956) finds a statistically significant difference (p < .01) in the underlying distributions of the data at the different levels of hearing impairment . However, the differences are small and are not believed to invalidate the global description of the data presented here . (It should also be noted that the data contain a complex mixture of independent and dependent observations ; the true p value is, Loudness Scaling Revisited/Elberling are used for the setting of a nonlinear hearing aid, the different procedures result in procedural effects-or biases-that will influence both the position and the slope of the 1/0 function . therefore, higher than . 01) . Second, it is important to acknowledge that, in this presentation, only the slope of the loudness function has been used . However, depending on the complexity of the underlying mathematical model, loudness functions acquired by loudness scaling need at Effect of Variance of Inverse Slope least one more parameter for their descriptionfor instance, slope and offset . For loudness restoration, the offset parameter controls the gain independent of input level and would, therefore, correspond to the setting of a linear hearing aid . Therefore, other parameters than slope have been ignored in the present evaluation . As evaluated for the normative data and for the hearing-impaired data, it appears to a first-order approximation that the variance of inverse slope is independent of the hearing loss, including normal hearing . Figure 9 gives : G50 = 50* (1/S- - 1/Sn) + HTL. G8o = 80* (1/Si -1/Sn) + HTL. Synthesis (2) (3) which inserted into formula 1 gives the following formulation for the compression ratio: Effect of Procedural Differences For the normative references evaluated earlier, the scaling procedures produce different relationships between the loudness categories and the acoustic (physical) domain (see Fig. 1 and Table 1) . Therefore, if an amplification strategy is targeted at a specific loudness level in normalhearing subjects (e .g ., comfortable), this would correspond to quite different acoustic sound levels in the hearing aid, dependent on the applied scaling procedure . Regarding the data from the hearing impaired, the different procedures yield different normalized data (see Table 3), which have yet to be related to acoustic settings of the hearing aid . Therefore, the information about the average relationship between inverse slope and hearing loss for each of the four procedures has been used to calculate the relative gain settings in a nonlinear hearing aid with linear WDRC . These gain settings are given in Table 4 as dif- 1/CRi = 1 + WS . - 1/Sn) (4) Now, for a given hearing loss, HTL., only 1/S; and 1/S n will be the unknown variables . Until now, we have assumed that loudness restoration means the restoring of loudness to its average normal value as represented by the normative reference. However, due to the significant variance in loudness data in normalhearing subjects, (see Table 2 and Fig. 7), it could be more appropriate to define loudness restoration as the restoration of loudness to the value for the individual hearing impaired had he/she been normal hearing. A consequence of not accepting this (new) definition would be that even subjects with normal hearing would require loudness restoration whenever their individual loudness function deviates from the normative reference! As described in formula 4, the problem is that there are two unknown quantities, which both contribute to the total variance, VARwCR>> of the target CR . Since it is the difference between the two variables that enters formula 4, the total variance can be developed, informally, as follows: ferences between the gain for high (80 dB SPL) and for low (50 dB SPL) input levels . The difference between the four procedures increases with the magnitude of the hearing loss . The maximum difference is 2.3 dB at 40 dB HL but amounts to 4.5 dB at 80 dB HL . Consequently, when measured loudness scaling data Table 4 Differences in Gain between a 80 dB HL and a 50 dB HL Input Sound Level for a Nonlinear Hearing Aid Fitted Based on the Results from the Four Procedures Shown in Figure 5A-D G,, -G,, (dB) at.: Elberling and Nielsen (1993) 40 dB HL -10 .9 80 dB HL -21 .8 60 dB HL -16.4 Kiessling (1995) -8 .6 -13 .0 -17 .3 Launer et al Ricketts and (1996) Gentler (1996) -9 .5 -10 .1 -19 .0 -20 .2 -14.2 -15.1 The average loudness scaling data (see Table 3) are used for the calculations . 257 Journal of the American Academy of Audiology/Volume 10, Number 5, May 1999 Table 5 Resulting Variance of the Estimated Target Compression Ratio Depending on the Fitting Alternative (A or B) and the Value of the Correlation Coefficient, r r= O r=0.5 r=1 .0 VAR(I)S) VAR(1/S) VAR(1/S) (A) with loudness scaling (B) without loudness scaling 2* VAR(1/S) VAR(1/S) 0 Sound Level (dB HL) too , lio Figure 9 Loudness model for normal and impaired hearing as shown previously in Figure 4. Here, the two sets of dotted lines indicate the variance of the slope of the loudness function : (left) S. in subjects with normal hearing and (right) S; in subjects with a hearing threshold = HTL. . Geo and Gso indicate the target gain necessary for restoring loudness at 80 and 50 dB HL, respectively. For further details, see text. VAR(VCR) _ (1isi)+VAR(vsa)- 2*COVvsO/sn)(5) where the COV term refers to the covariance between the observed distribution of inverse slope for subjects with a given hearing loss, HTL;, and the corresponding distribution for the same subjects had they been normal hearing. Since VARaisO and VAR(1/sn) are approximately the same (= VAR(vs), as found in the section on variation of inverse slope across hearing-impaired subjects and Fig. 7) and the covariance can be expressed by the correlation coefficient, r, by rais(,lisn)=COV(vs(,vsn)/ [VAR(vs1) VAR(lisn)7 (6) we arrive at the following expression for the total variance : VAR(1/CR)=2 * VAR(vs)* [l-rwsOvsn)j (7) Now, let us consider the alternatives : (A) individual loudness is measured by loudness scaling and used together with the normative reference for fitting of the hearing aid's CR, and (B) individual loudness is not measured but the estimated value of 1/S; is used together with an individual, unknown reference for the fitting. 258 Alternative A. If we disregard any test-retest variability and other uncertainties in the acquisition of the individual loudness data, both the variance, VAR(1/si) and the COV term in equation 5 will be zero . The total variance VAR(1/CR) will be constant = VAR(g) . Alternative B. The total variance, VAR(1,CR), will depend upon the correlation coefficient, raisijisn), in equation 7. The results corresponding to three different values of r are indicated in Table 5. What do we know about the correlation coefficient r(vSOisn)? Unfortunately, it is only on rare occasions that loudness information would be available for the individual subject both with and without a hearing impairment. However, a positive correlation possibly exists because we expect, intuitively, that both the perception and the interpretation of different loudness categories are subject specific . In an attempt to obtain more information about this correlation, the following analyses were carried out: 1. The scaling data by Elberling and Nielsen (1993) contain loudness data from 29 subjects at 500 Hz and 2 kHz. The inverse slope values were corrected for the mean effect of the hearing loss by calculating the difference between each value and the linear trend given by 1/S = -0 .0091*HTL + 1 .0 (see Table 3) . Between the corrected inverse slope values at 500 Hz and at 2 kHz, a correlation coefficient of 0.38 was found (t = 2.15, df = 27, p < .02). 2. Similarly, the scaling data by Kiessling (1995) contain loudness data from 116 subjects measured both at 500 Hz and 4 kHz. As above, correction was made for the mean effect of the hearing loss by calculating the difference between each value and the linear trend given by 1/S = -0 .0072*HTL + 1.0 (see Table 3) . Between the corrected values at the two frequencies, a correlation coefficient of 0 .42 was found (t = 4.9, df = 114, p < .0005) . Loudness Scaling Revisited/Elberling 3 . In the same set of data by Messling (1995), it was possible to isolate 70 subjects with sloping hearing losses (HTL4kHz - HTL50OHz > 15 dB HL) . The average hearing loss was 26 dB HL at 500 Hz and 65 dB HL at 4 kHz, which means close to normal hearing at low frequencies and a substantial hearing loss at high frequencies . As above, the inverse slope values were corrected for the mean effect of the hearing loss and a correlation coefficient of 0 .41 was found (t = 3 .7, df = 68, p < .001) . This analysis expresses the relationship between the loudness in frequency areas corresponding to normal and to impaired hearing when the effect of the hearing loss has been removed . The above analyses demonstrate that the loudness scaling values obtained at different frequencies in the individual are positively correlated-also when comparing frequency areas with approximately normal and impaired hearing . The results support the underlying assumption for the alternate monaural loudness balance test (Reger, 1936), which has been used extensively for the diagnostic evaluation of recruitment in bilateral hearing losses . Since most loudness scaling studies both in subjects with normal hearing and hearing impairment have indicated a significant effect of hearing loss but only a limited (if any) effect of frequency, the results also indicate the expected positive correlation between the loudness scaling values obtained in the individual hearing impaired with and (hypothetically) without a hearing loss . If in equation 7 we substitute the unknown correlation coefficient with the value =0 .4, the total variance of the calculated compression ratio becomes =1 .2*VARn/s) - VAR,l/s) . For all practical purposes, this surprising result means that the variance of the target CR would be independent of whether loudness scaling is performed or not! SUMMARY AND CONCLUSIONS 1 . In normal-hearing subjects, the different scaling procedures relate the perceptual categories differently to the physical domain (viz . sound level) . 2. Normal-hearing subjects display a large variance in the sound level associated with a specific loudness category (approximately 35 dB at the comfortable level) . 3 . In about 70 percent of the hearing impaired, the slope of the measured loudness function can be predicted from the hearing loss with an accuracy corresponding to a ±5-dB finetuning of gain . 4. The remaining 30 percent of hearingimpaired persons could be referred to as outliers . These outliers consist of the "sound sensitive" (12%) and the "sound addicts" (17%). The latter could also reflect an inhomogeneous diagnostic classification . 5 . When individual loudness data are used to fit a nonlinear hearing aid, the scaling procedures will produce different I/O characteristics of the hearing aid . 6 . If a broader view is made on the use of loudness scaling for fitting nonlinear hearing aids, it becomes apparent that the statistical properties of the normative reference are as important as the variance of the loudness functions in the hearing-impaired subjects . Therefore, the variance of the setting of the CR in a nonlinear hearing aid seems to be independent of whether the loudness function is measured or estimated . It should be stressed that, for the reasons given in the introduction, the loudness dimension is important both for the design and the fitting of modern hearing aid algorithms . However, the multiple effects described above make it difficult to promote the general use of loudness scaling procedures for the fitting of nonlinear hearing aids . Acknowledgment. I would like to thank the authors of the following publications : Kiessling (1995), Launer et al (1996), and Ricketts and Bentler (1996) for providing their original data from their hearing-impaired subjects . 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