Adaptive Procedures for Hearing Aid Prescription and Other Audiologic Applications

J Am Acad Audiol 3: 119-131 (1992) Adaptive Procedures for Hearing Aid Prescription and Other Audiologic Applications Harry Levitt* Abstract A class of adaptive procedures is described. These procedures are relatively easy to implement and have many useful properties for audiologic applications. Recent advances in programmable hearing aids enable these techniques to be used effectively in adaptive prescriptive fitting of a hearing aid. All of the procedures are based on the simple up-down technique that, in addition to its simplicity, is both efficient and reliable. Key Words: Hearing aids, adaptive procedures, hearing aid fitting n adaptive test procedure is one in w hii ch the experimental conditions for each measurement are determined by the results of previous measurements. A particularly important class of adaptive procedures is based on the simple up-down technique in which the stimulus level on any given trial is determined by the subject's response on the preceding trial. For example, in measuring the auditory threshold using this technique, the stimulus level is decreased following a trial in which the subject reports that the stimulus is audible. If the subject reports that the stimulus is inaudible, then the stimulus level is increased on the next trial. Stimulus levels chosen in this way will oscillate about a mean level at which the stimulus is audible 50 percent of the time. Variations of the simple up-down procedure have been used intuitively for well over a century for a variety of applications. These include such diverse applications as docking a ship, controlling the speed of a steam engine, or testing explosives, not to mention its many applications in the behavioral sciences. It was only fairly recently, however, that the statisti- *Center for Research in Speech and Hearing Sciences, Graduate School and University Center, City University of New York Reprint requests : Harry Levitt, Center for Research in Speech and Hearing Sciences Graduate School and University Center, City University of New York, 33 West 42nd Street, New York, NY 10036 cal properties of the simple up-down technique were analyzed in any depth (e.g., Anderson et al, 1946 ; Dixon and Mood, 1948 ; Robbins and Monro, 1951). These analytic treatments not only showed that the simple up-down technique is an extremely efficient procedure (for certain applications), but also provided rules for implementing the technique so as to maximize efficiency and reduce estimation bias. These rules relate to the choice of step size, the stimulus level to be used on the first trial, and various methods of estimating the 50-percent stimulus level (the stimulus level yielding responses of a prescribed type 50 percent of the time). The first significant application of the simple up-down technique in audiology, as refined by the above-mentioned analytic treatments, was that of Jerger et al (1959) in which the auditory thresholds for pure tones and speech were measured using the same precise procedure. Jerger et al followed the specific recommendations of Dixon and Massey (1951) in implementing the procedure. It is important to bear in mind that although the technique is simple to implement, variations in its implementation can produce substantial changes in the resulting estimates. It is therefore important to choose a version of the technique that provides the desired information efficiently and reliably, and to use that version of the technique consistently across experimental conditions. The simple up-down technique is particularly well suited for estimating the 50-percent 119

Journal of the American Academy of Audiology/Volume 3, Number 2, March 1992 stimulus level on a performance-intensity function (or any response curve that increases monotonically with stimulus level). It is not a good procedure for estimating other parameters of a response curve, such as the slope, or stimulus levels other than the 50-percent level. As a consequence, variations of the simple updown technique have been developed in order to estimate parameters other than the 50-percent level of a response curve. Adaptive techniques of the up-down type are, by design, relatively easy to implement. The statistical properties of these techniques, however, are not easy to analyze. Theoretical analyses of up-down techniques have been carried out for experiments involving few trials and for experiments involving many trials. However, for experiments involving a moderate number of trials, as typically occurs in audiology, the theoretical analysis is extremely complex. As a consequence, statisticians have used computer simulations of these techniques to determine their statistical properties (Wetherill, 1963, 1966). Computer simulation is a powerful method of analysis, but it is also a relatively time consuming and cumbersome approach in comparison with a general theoretical treatment. The simple up-down technique has been studied in great detail using both theoretical and computer-simulation methods of analysis. The statistical properties of more advanced variations of the up-down technique, however, have yet to be analyzed. In order to address this problem, a family of up-down test procedures has been developed having the same general properties as the simple up-down technique (Wetherill and Levitt, 1965 ; Levitt, 1971). In this way, the properties of each adaptive technique within this family of adaptive procedures can be analyzed by analogy with the simple updown technique. All of the analytic and empirical results already obtained in evaluating the simple up-down technique can thus be used in determining the properties of each of these more sophisticated up-down procedures. The above process of generalization led to the development of the so-called transformed up-down procedures. Application of these techniques in audiology led to the development of new variations of the transformed up-down procedure, including the development of procedures for estimating the peak of a performanceintensity function exhibiting rollover and for the prescriptive fitting of hearing aids (Levitt, 1978a, b ; Neuman et al, 1987). This paper reviews the simple up-down technique and provides rules for its implementation. This is followed by a description of how the technique has been generalized (the transformed up-down procedure, multivariate pairedcomparison testing) and how these more general versions of the up-down technique have been applied to problems in audiology including hearing-aid prescription. SIMPLE UP-DOWN TECHNIQUE T he simple up-down technique forms the basis of all the adaptive procedures to be discussed, and it is important to begin with a description of the technique and its implementation. Figure 1 shows a typical set of data. The horizontal axis shows trial number and run number (to be defined later). The vertical axis shows stimulus level. Arbitrary units are used for purposes of illustration. In practice, stimulus levels are likely to be specified in db SPL, db HL, or db signal-to-noise ratio, depending on the type of test that is administered. The stimulus level used on the first trial is known as the initial value. In this example, the test began with an initial value of 0 units. The data show that a positive response was obtained, on this first trial, as indicated by the + symbol. The definition of a positive response depends on the nature of the test. For example, a + response could be defined as a "signal audible" response in a test designed to measure the auditory threshold, or it could be the correct I 2 3 4 5 6 Run Number Figure 1 Typical data record for the simple up-down technique. The stimulus level is decreased after every positive response and increased after every negative response. A sequence of changes in one direction only is defined as a run. (Figure reproduced from Levitt, 1971.) 120

Procedures for Hearing Aid Prescription/Levitt identification of the test word in a speech recognition test. The rule used in the simple up-down technique is to decrease stimulus level after a positive response, and to increase level after a negative response (e.g., stimulus inaudible, or test word not identified correctly). Since a + response was obtained on Trial 1, the stimulus is reduced in level for the next trial. The amount by which the stimulus level is reduced is known as the step size. In this example, the step size is 1 unit and the stimulus level on Trial 2 is, therefore, -1 units. The data show that a positive response was also obtained on Trial 2, and as a consequence, the stimulus level for Trial 3 is reduced by one more step to -2 units. On this trial, however, a negative response was obtained, and accordingly, the stimulus level was increased by one step to -1 units for Trial 4. Since a change has occurred in the direction of adjusting stimulus level, this point in the data record is referred to as a reversal. Negative responses were obtained on Trials 3, 4, 5, and 6, and as shown in the diagram, stimulus level is increased by one step after each of these negative responses. A positive response was obtained on Trial 7 leading to a reduction in stimulus level on the next trial. Trial 7 thus corresponds to the second reversal in the data record. For reasons that are obvious from the diagram, the reversal on Trial 3 is defined as a valley, and the reversal on Trial 7 is defined as a peak. The sequence of trials leading up to a reversal is defined as a run. As shown in the diagram, Run 1 is made up of Trials 1, 2, and 3. The second run is made up of Trials 4, 5, 6, and 7. The simple up-down technique is designed to converge on the stimulus level corresponding to 50-percent positive responses. There are several ways in which the 50-percent stimulus level can be estimated. A relatively simple method with good statistical properties is to take the average of successive pairs of reversals (i.e., an adjacent peak and valley) as estimates of the 50-percent stimulus level. Since there may be some adaptation at the start of a test, it is often recommended that the first reversal be omitted from the data analysis. Using the above procedure, the first estimate of the 50-percent stimulus level is the average of the peak occurring on Trial 7 (= second reversal) and the valley occurring on Trial 9 (= third reversal). The average stimulus level corresponding to these two reversals is [2 + 01/2 = 1 units. A second estimate of the 50-percent stimulus level is obtained by taking the average of the next peak and valley in the data record. These reversals occur on Trials 12 and 17, respectively, yielding an estimate of [3 + (-2)]/2 = 0.5 units for the 50-percent stimulus level. A third estimate is obtained by taking the average of the peak and valley occurring on Trials 20 and 22, respectively. This third estimate has a value of [ 1 + (-1)1/2 = 0 units. Unless there is evidence that the 50-percent stimulus level is changing as the test progresses, it is common practice to take the average of these three estimates, [1 + 0.5 + 01/3 = 0.5 units, in order to obtain a more precise estimate of the 50-percent stimulus level. Note that the three estimates are not statistically independent of each other. This is a factor to be taken into consideration if within-test variability is to be estimated from the data (Levitt, 1978a). Four important decisions need to be made by the experimenter in using the simple updown technique. These are : (1) the choice of initial value, (2) the choice of step size, (3) when to terminate testing, and (4) choosing an appropriate method of estimation. Simple, practical rules for making these decisions, and their underlying rationale, are presented in the paragraphs that follow. Rule 1 The initial value should be close to the stimulus level being estimated (e.g., the 50- percent stimulus level for the simple up-down technique). For maximum statistical efficiency, the initial value should be the best available estimate of the 50-percent stimulus level. On the other hand, it may be helpful for the subject to begin with a stimulus that is relatively easy to recognize on the first trial. For this reason, an initial value above the 50-percent stimulus level is often used. A situation to avoid, however, is that in which the initial value is many steps above the 50-percent stimulus level. If this occurs, many trials providing little information may be used up on the first run. Rule 2 The step size should be approximately one third to one sixth of the transition region of the response curve. For practical purposes, the transition region of the response curve can be defined as the range from the 10-percent stimulus level to the 90-percent stimulus level. For 121

Journal of the American Academy of Audiology/Volume 3, Number 2, March 1992 example, if the transition region is 6 db wide, then a good choice of step size is 1 to 2 db. If the response curve is asymmetric (e.g., very steep above the 50-percent stimulus level and shallowbelow this level), then a step size that is just a little large for the steep half of the transition region should be chosen. An inappropriate choice of step size will affect the adaptive procedure in one of two ways. A very small step size will increase precision of estimation slightly, but at the possible cost of a long and wasteful first run if the choice ofinitial value is poor. Avery large step size will protect against a poor choice of initial value, but will result in a less precise estimate of the targeted stimulus level. A practical compromise is to start with a large step size to guard against a poor choice of initial value and then to reduce the step size after the first reversal. A statistically efficient rule for adjusting step size as the test progresses has been developed by Robbins and Monro (1951). A simplified approximation to the Robbins and Monro rule that is relatively efficient is to halve the step size after the first reversal, halve again after the third reversal, and if the test continues for well over eight runs, halve again after the seventh reversal. Rule 3 Testing should be terminated once the targeted stimulus level has been estimated with the required precision. If the response curve does not change with time, then, in general, the greater the number of trials the more precise the resulting estimate. A common limitation, in practice, is the time available for testing. Typically, data need to be obtained in as short a period of time as is practical. The shortest meaningful test of the up-down type is that in which the test is terminated after only one reversal. The resulting estimate, however, is likely to be biased and of low precision. A much better estimate is obtained if testing continues for at least two more reversals (i.e., testing is terminated after the third reversal). If time permits, it is recommended that testing continue for at least seven reversals (as shown in Figure 1). A second factor limiting precision of estimation is the stability ofthe subject's responses. This may be an important factor for subjects who have never been tested before, or for tests that continue for long periods of time. A simple check on the stability of the subject's responses is the variability of the estimates obtained from successive pairs of reversals. If the estimates show a statistically significant trend over time, then the subject may not have adjusted to the test conditions. A fairly common situation is that of an apparent trend during the early part of a test followed by relatively stable estimates for the remainder of the test. An apparent trend of this type can result statistically from a poor choice of initial value. Whatever the cause (statistical or behavioral), data from only the latter part of the test should be used in estimating the targeted stimulus level. Rule 4 There are several possible rules for estimating the targeted stimulus level from data of the up-down type. The method of averaging successive pairs ofreversals (i.e., adjacent peaks and valleys) is recommended since the technique is simple and relatively efficient (Wetheril1,1963). This method of estimation also does not require that a specific mathematical form be assumed for the response curve (i.e., the technique is nonparametric). An additional advantage is that a series of estimates can be obtained as the test progresses, thereby allowing the experimenter to check if the subject's responses are stable over time. More sophisticated methods of estimation such as Probit analysis (Finney, 1952) can also be used, but the increase in precision of estimation for samples of the size typically encountered in audiology is likely to be small (Levitt, 1971,1978a). These more sophisticated estimation procedures can be used for obtaining supplemental information, such as estimating the slope of the response curve in addition to the 50- percent stimulus level. It is important to bear in mind, however, that the simple up-down technique effectively concentrates data in the vicinity of the 50-percent stimulus level and that data obtained in this way are poorly suited for estimating other parameters of the response curve, such as the slope or stimulus levels far removed from the 50-percent stimulus level. The preceding comments highlight a common problem, that of relying on sophisticated methods of data analysis to remedy the problems introduced by poor experimental design. If, for example, an experimenter is interested in measuring the slope of a response curve, then an experimental procedure that is designed to place observations in the appropriate region of the response curve for estimating slope should 122

Procedures for Hearing Aid Prescription/Levitt be used (adaptive techniques for this purpose are described by Levitt, 1971). It is poor design practice to use a technique that is well suited for one application in the hope that it will also work well in other applications. This caution is particularly relevant to adaptive techniques designed for a specific application since these techniques are usually highly efficient in their intended application but can also be highly inefficient in applications for which they were not intended. Adaptive techniques have much to offer the audiologist, but it is important to choose the right procedure for the task at hand. For example, the simple up-down technique is well suited for estimating the auditory threshold using responses of the type "signal is audible" and "signal is inaudible." If, however, a two-alternative forced-choice technique is used, 50-percent positive responses can be obtained by random guessing alone and an adaptive procedure for estimating a point on the response curve well above the 50-percent stimulus level should be used. In another application, it may be of interest to determine the width of a performanceintensity (PI) function. In this case, two points spaced fairly far apart on the PI function should be estimated. The following section deals with the problem of estimating points other than the 50-percent stimulus level. TRANSFORMED UP-DOWN PROCEDURE he simple up-down technique is particularly well suited for estimating the 50- T percent stimulus level on a response curve. It is not well suited for estimating other points on a response curve. Adaptive rules for estimating various other points on a response curve are shown in Table 1. Perhaps the most well known of these rules is the "2-down 1-up rule" (entry 2 in the table) for estimating the 70.7-percent stimulus level. In this case, the stimulus level is reduced after two consecutive positive responses (i.e., after a response sequence of the form ++). In contrast, stimulus level is increased immediately after a negative response (i.e., after either a - or a +- response sequence). As noted earlier, considerable effort has been invested in analyzing the statistical properties of the simple up-down technique. This effort has been useful in that simple rules have been developed, such as those described in the preceding section, for the efficient implementation of the simple up-down technique in practice. It would be helpful if these analyses could also be used in determining the statistical properties of other adaptive procedures of the updown type. This can be done by redefining what is meant by a positive response. Consider, for example, entry 2 in Table 1. If the response sequence, ++, is defined as a transformed positive response (#) and either of the response sequences, - or + - is defined as a transformed negative response (=), then the 2- down 1-up rule is reduced to the simple updown rule in terms ofthe transformed responses. That is, stimulus level is decreased after a transformed # response (i.e., after a ++ response sequence) and increased after a transformed = response (i.e., after either a - or +- response sequence). Table 1 Transformed Up-Down Procedures for Estimating Points Other Than the 50-Percent Stimulus Level Entry Number Response Sequences for Response Sequences for Transformed Transformed Positive Response, # Negative Response, = Percent Positive (+) Responses at Convergence 7 + 29.3 2 70.7 3 20.6 4 +++ 79.4

Journal of the American Academy of Audiology/Volume 3, Number 2, March 1992 a -c J Z) N>1-3 +" -1 `- -2-3 0 10 20 30 40 a a a a a a a = a a a = L,i,I, i -~I'I, I~,I1 i 10 20 30 40 Trial Number I 2 3 4 S 6 L 7 ~ Run Number Figure 2 Typical data record for the 2-down 1-up transformed up-down procedure. The upper half of the figure shows a typical data record for the transformed updown procedure designed to converge on the 70.7-percent stimulus level (entry 2 of Table 1). The stimulus level is decreased after a ++ response sequence and increased after either a -or +- response sequence. The lower half of the figure shows the same data record ifevery ++ response sequence is replaced by the # symbol for a transformed positive response and every- or +- response sequence by the = symbol for a transformed negative response. The transformed data record is analogous to that for the simple up-down technique, as shown in Figure 1. (Figure reproduced from Levitt, 1978a.) The upper half of Figure 2 shows a typical data record for the 2-down 1-up adaptive procedure. The lower half of the diagram shows the same set of data using the transformed response symbols (# and =). Note that the data record for the transformed response symbols is identical to that for the simple up-down technique shown in Figure 1. This has been done for purposes of illustration. The point being made is that the mechanics of the 2-down 1-up adaptive procedure reduces to that of the simple updown technique by means of a simple transformation (i.e., by introducing the concept oftransformed positive (#) and transformed negative (=) responses, fairly complex adaptive procedures can be reduced in form to that of the simple up-down technique). The above transformation has many useful properties. It is both simple and efficient. In addition, the statistical characteristics of the transformed up-down procedure can be deter- mined directly from the known characteristics of the simple up-down technique as already determined from extensive theoretical analyses and computer simulations. Further, the rules developed for the efficient implementation ofthe simple up-down technique can also be applied directly to the transformed up-down procedure. The rules for implementing a transformed up-down procedure are described most conveniently in terms of the transformed response curve. The upper curve in Figure 3 is the subject's response curve. This curve shows the percent of positive responses (+) as a function of stimulus level. The lower curve is the transformed response curve showing the percent of transformed positive responses (#) as a function of stimulus level. This particular transformed response curve is for the 2-down 1-up adaptive procedure (entry 2 in Table 1). Whereas the simple up-down technique converges on the 50- percent stimulus level of the subject's response curve, the transformed up-down technique converges on the 50-percent response level of the transformed response curve. As shown in the figure, the 50-percent stimulus level of the transformed response curve corresponds to'the 70.7-percent stimulus level of the subject's response curve (i.e., this particular transformed up-down technique will provide an estimate of the stimulus level corresponding to 70.7 positive (+) responses by the subject). As before, the first rule is to choose an initial value close to the targeted stimulus level. In this case, the targeted level is the 50- percent stimulus level of the transformed response curve that, in turn, corresponds to the subject's 70.7-percent stimulus level. If the experimenter does not have even a rough idea of Figure 3 Transformed response curve compared to subject's response curve. The upper curve is the subject's response curve. The lower curve is the transformed response curve for the 2-down 1-up adaptive procedure.

Procedures for Hearing Aid Prescription/Levitt where the 50-percent stimulus level is located, or if the subject needs to be provided with an easily recognized stimulus at the start of the test, then it is advisable to begin with an initial value that is well above the transition region of the subject's response curve and proceed in relatively large steps using the simple up-down technique until the first reversal is reached. This strategy should locate the transition region of the subject's response curve with only a few trials. Since the transition region of the transformed response curve overlaps that of the subject's response curve, this strategy also works well in locating the transition region of the transformed response curve. Once the first reversal has been reached, the transformed updown rule should be used on subsequent trials with a step size that is appropriate for the transformed response curve. The step size should be approximately one third to one sixth the width of the transformed response curve. The width of the transformed response curve is defined here as the range from the 10-percent stimulus level to the 90-percent stimulus level. If the transformed response curve does not fall below the 10-percent response level, as can occur in a forced-choice test with a limited number of alternatives, then, for the purpose of choosing an appropriate step size, the width of the transformed response curve may be defined as twice the stimulus range from the 50-percent to 90-percent stimulus levels. Note that the width of the transformed response curve shown in Figure 3 is only slightly less (about 20 percent less) than that of the subject's response curve and, consequently, since the appropriate choice of step size lies within a fairly broad range, the same step size could be used for both the simple up-down technique and this particular transformed updown procedure. The rules for terminating the test and estimating the targeted stimulus level are exactly the same as those for the simple up-down technique. Specifically, it is recommended that testing continue for at least seven reversals, and that the last six reversals be used in obtaining the estimate. In the case of a poor choice for the initial value (e.g., if the initial value is very different from the final estimate), or if there is evidence that the subject is still adapting to the task after the first reversal, then testing should continue for several more reversals. As before, it is recommended that only the last six reversals be used in obtaining the estimate. TRANSFORMED UP-DOWN PROCEDURE FOR ESTIMATING PB-mAx key assumption underlying any adaptive A technique of the up-down type is that an increase in stimulus level is always accompanied by an increase in the probability of a positive (or transformed positive) response. Expressed mathematically, the response curve is assumed to vary monotonically with stimulus level. This is not always the case, and special care must be taken with response curves showing rollover. A simple precaution is to restrict the choice of stimulus levels to only one side of the peak in the response curve. In many cases, however, the audiologist's interest lies in finding the peak ofthe response curve (e.g., estimating PB-max in a performance-intensity function showing rollover). Under these conditions, a different type of transformed response is necessary. An adaptive procedure for estimating PBmax is based on estimating the slope of the performance-intensity (PI) function. The reason for doing this is that the slope of the PI function varies monotonically with stimulus level in the vicinity of PB-max, thereby satisfying the basic requirement for using an adaptive procedure of the up-down type. The upper half of Figure 4 shows a PI function exhibiting rollover. Note that the PI function to the left of the peak has an upward (positive) slope and a downward (negative) slope to the right of the peak (i.e., the slope decreases systematically from a positive value to a negative value as stimulus level is increased from below PB-max to above PB-max). At the stimulus level corresponding to PB-max, the slope is zero. An adaptive procedure that estimates slope and which converges on the stimulus level corresponding to zero slope would, in this context, provide a means for estimating PB-max. The simplest possible estimate of slope requires two observations, one at each of two relatively closely spaced points on the PI function. There are four possible outcomes for this pair of measurements, ±, +, +, and -, where the upper symbol indicates the response at the higher stimulus level and the lower symbol represents the response at the lower stimulus level. The response pair +, which is indicative of a negative slope, is defined as a transformed positive response, #. Similarly, the response pair ±, which is indicative of a positive slope, is 125

Journal of the American Academy of Audiology/volume 3, Number 2, March 1992 U 0 U. CZ O C am c a`> -a LL- 100 -o 80 E L C/5 c O O a e0 "Cr H C>. 40 N i N a o a- 20 0 1 Pe Max 1 1 1 1 I i 1 I I 60 70 80 90 100 110 Stimulus Level (db SPL) 1 1 I A I I I 60 70 80 90 100 110 Mean Stimulus Level (db SPL) Figure 4 Performance-intensity function exhibiting rollover and corresponding transformed response curve. The upper half of the figure shows a typical performanceintensity (Pl) function exhibiting rollover. The test score at the peak of the curve is defined as PB-max. The lower half of the figure shows the resulting transformed response curve for an adaptive procedure in which pairs of stimuli spaced 10 db apart have been used in estimating the slope of the PI function. (Figure reproduced from Levitt, 1978a.) defined as a transformed negative response, _. The response pairs +and =provide no information on slope and are treated as ambiguous responses requiring a repetition of the stimulus pair before a decision can be made regarding a change in stimulus level. The lower half of Figure 4 shows the transformed response curve corresponding to the PI function in the upper half of the figure. Note that the probability of a transformed positive response increases monotonically with stimulus level and that the 50-percent stimulus level of the transformed response curve corresponds to PB-max (i.e., the transformed up-down pro- cedure will converge on that stimulus level yielding PB-max). The transformed response curve in this example is not symmetric about the 50-percent stimulus level (i.e., the curve is less steep at stimulus levels below PB-max and appears to flatten out altogether at the lowest stimulus levels). This could cause problems for the adaptive procedure at these low stimulus levels. A long sequence of - response pairs would be indicative that the stimulus levels being used are well below PB-max. It should be noted that PB-max occurs at a relatively high stimulus level and that the subject's threshold of discomfort is less than two step sizes above the stimulus level for PB-max. Stimulus levels above the threshold of discomfort should not be used. A simple rule for protecting the subject when the adaptive procedure calls for the use of an unacceptably high stimulus level is to skip that particular trial and to assume that, had such a high stimulus level been used, a transformed positive response would have been obtained, thereby requiring a lower stimulus level on the next trial. In this way, the threshold of discomfort serves as an upper boundary above which the adaptive procedure cannot go. A not uncommon result is for the adaptive procedure to keep reaching this upper boundary, indicating that PB-max lies at the threshold of discomfort or above. A typical data record is shown in Figure 5. The data show the classic up-down pattern of responses and the stimulus level corresponding to PB-max can be estimated by taking the average of successive pairs of reversals. As before, the same set of rules applies in terms of implementing the procedure efficiently. The initial value should be close to the targeted value. Since PB-max is likely to fall close to the threshold of discomfort, a good choice for the first pair of stimuli would be just below this threshold. The ideal step size is a little harder to determine since the transformed response curve is asymmetric. For a typical PI-function occupying a range of 30 to 40 db, a step size of one quarter this range (10 db) appears quite reasonable. For convenience, the spacing between pairs of observations can also be made equal to this step size. The rules for terminating the test and estimating the stimulus level corresponding to PBmax can be applied without difficulty. Referring to Figure 5, the stimulus levels corresponding to the last six reversals in the data record are 80, 100, 80, 90, 80, and 100 db SPL, respect- 126

II II uili luuinn Inui u'heii Procedures for Hearing Aid Prescription/Levitt 110 100 J J r v) 90 7 771 m E _S~ 80 W T0 F ++ 7++ +/+ - +t+- ++/+ PROPORTION 1y CORRECT 13 0.33 15 0.80 19 0. T9 7 0.14 60 1 1 LL I ( 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1. 1 1 3 5 7 9 19 29 '.b 49 2 4 6 9 10!0 30 40 30 Trial Number I 1 I _. 1 1 I Run Number : I z 3 4 5 s 7 Figure 5 Typical data record obtained in estimating PB-max. The data record is for the transformed up-down procedure designed to converge on 1`13-max using paired stimuli. The mean stimulus level for each stimulus pair is shown by the solid circle. The response to each stimulus within a pair is shown by a positive or negative symbol, depending on the response obtained. One stimulus in the pair is always 5 db below the mean stimulus level, the other 5 db above the mean, the order of presentation within a pair being randomized. The placement of the mean stimulus levels is analogous to the placement of observations in the simple up-down method. The lower section of the horizontal axis shows the runs. The two columns on the right-hand side of the figure show the number of stimuli and the proportion of positive responses, respectively, obtained at each stimulus level used in the test. (Figure reproduced from Levitt, 1978a.) ively. The average of these peaks and valleys is 88.3 db SPL (i.e., the stimulus level at which PB-max occurs is estimated to be 88.3 db SPL). This estimate is reasonably close to the true value of 90 db SPL. It is also evident from the shape of the response curve (upper half of Fig. 4) that the speech test score measured at a level of 88.3 db SPL is unlikely to differ significantly from the true value of PB-max for any of the standard tests used in speech audiometry. A rough estimate of the value of PB-max can be obtained by finding the percent of correct responses for trials at the stimulus level that is estimated to yield PB-max. It is recommended, however, that a more reliable estimate of PBmax be obtained by repeating a full speech identification test at this estimated stimulus level. ADAPTIVE STRATEGIES FOR HEARING AID PRESCRIPTION T he problem of adjusting a hearing aid for maximizing some desirable property of the amplified signal (e.g., intelligibility, sound quality) is conceptually similar to that of finding the peak of a response curve except that the problem is much more difficult since more than one variable needs to be adjusted. In the case of estimating PB-max, only one variable, stimulus level, was adjusted adaptively. For a hear- ing aid, several variables may need to be adjusted, such as the gain in different frequency regions. In the case of a modern compression hearing aid, additional variables such as compression ratio, compression threshold, attack time, and release time may also need to be optimized for each hearing aid user. The problem is even more difficult in that the optimum hearing-aid settings for one acoustic environment (e.g., a quiet living room) may not be the same as for another acoustic environment, such as a busy office. A third complicating factor is that the optimum hearing aid settings may also differ depending on the user's criterion for optimality (e.g., hearing-aid settings for maximum speech intelligibility may not necessarily be the same as those for maximizing user comfort or overall sound quality). Although the problem is very difficult, it is not beyond a practical solution. Multivariate adaptive procedures, generalized from the simple up-down technique, can be used for this purpose. The instrumentation needed for implementation of these adaptive techniques has also recently become available with the introduction of programmable hearing aids. For purposes of illustration, a two-variable problem will be considered ; adjusting the lowfrequency and high-frequency gain for optimizing, to a first approximation, the frequencygain characteristic of a programmable hearing 127

Journal of the American Academy of Audiology/Volume 3, Number 2, March 1992 aid. This was the approach used by Neuman et al (1987) based on a method introduced by Levitt (1978b). A matrix of frequency-gain characteristics is shown in Figure 6. Each cell in the matrix is identified by a pair of coordinates, x and y. The x coordinate specifies the loudness level of the speech signal in the low-frequency range (100-800 Hz). The y coordinate specifies the loudness level in the high-frequency range (800-5000 Hz). For example, the Cell (3,5) identifies a frequency gain characteristic that would place the speech signal at loudness level 3 (average comfortable level) in the low frequencies and loudness level 5 (average loud level) in the high frequencies. The loudness levels used were derived using the loudness scaling procedure developed by Pascoe (1978). Growth of loudness with signal level for one-third octave bands of noise was first measured using a six-point categorical scale (very soft, soft, comfortable, loud, very loud, and too loud). These loudness ratings were then used to obtain the low-frequency and highfrequency gains in order to raise the long-term rms level of conversational speech to the loudness levels shown in Figure 6. For example, the 1 2 m U m 3 7 <T am u- 4 Loudness Levels in the High Frequencies 1 2 3 4 5 - II -IV III ~ II I IV '-V II I -~I 1. Highest Very Soft 2. Average Soft 3. Average Comfort 4. Highest Comfort 5. Average Good Figure 6 Example of the modified simplex procedure. The vertical axis shows loudness levels of the speech signals in the low frequency region, 100 to 800 Hz ; the horizontal axis shows loudness levels of the speech signal in the high frequency region, 800 to 5000 Hz. (Figure reproduced from Neuman et al, 1987.) I gain for loudness level 1 would place the rms level of speech at the highest level obtained for the very soft rating (for the given frequency range). The gain for loudness level 2 would place the rms level of speech signal at the average soft level, loudness level 3 at the average comfortable level, loudness level 4 at the highest comfortable level, and loudness level 5 at the average loud level. Loudness levels covering the very loud and too loud ratings could also be included in the matrix, but previous experience with this technique has shown that these relatively high loudness levels are seldom chosen. The problem, in the context of the matrix shown in Figure 6, is now reduced to finding that cell in the matrix that corresponds to a prescribed criterion of optimality. In this example, the criterion used was that of maximizing intelligibility for continuous discourse in a background of cafeteria noise (speech to noise ratio = 5 db). Paired comparison judgments of relative intelligibility were used to search for the frequency-gain characteristic that maximized relative intelligibility. The first stage in the procedure is to begin with a good first estimate of the optimum frequency-gain characteristic. Cell (3,3), which corresponds to speech at a comfortable loudness level in both the low and high frequencies, represents a reasonable choice for the initial value. The next step is to determine whether intelligibility can be improved by examining neighboring cells. The situation is not very different from that described in the preceding section in which speech recognition was measured for two points spaced one step apart on a PI-function in order to search for the peak of the function. In this case, however, the search involves two variables and hence the change in intelligibility has to be measured on two dimensions. This is done by comparing intelligibility between the initial estimate, cell (3,3), and a cell one step away in the horizontal direction, cell (3,2), and between the initial estimate and a cell one step away in the vertical direction, cell (2,3). The three cells used in this first set of comparisons, cells (3,3), (3,2), and (2,3), form an L-shaped elemental unit identified by the symbol I in Figure 6. Paired-comparison judgments of relative intelligibility along the horizontal axis (i.e., cell (3,3) compared to cell (3,2) show that cell (3,2) is judged to yield more intelligible speech). This result is shown by the asterisk on the horizontal leg of the L-shaped unit. Paired comparisons along the vertical axis (i.e., cell 128

Procedures for Hearing Aid Prescriphionlev1L1 (3,3) compared to cell (2,3) show that cell (3,3) is judged to yield more intelligible speech). A second L-shaped grouping of cells, elemental unit II in Figure 6, is chosen according to the simple up-down rule operating in two dimensions. Consider first the horizontal dimension. Of the two cells compared on the horizontal dimension, the cell on the left hand side, cell (3,2), was judged to yield more intelligible speech. It is thus possible that even greater intelligibility could be obtained by examining cells to the left of cell (3,2). Consequently, the horizontal leg of the L-shaped unit was moved to the left by one step so as to occupy cells (3,1) and (3,2), respectively. Consider now the vertical axis. Of the two cells compared on the vertical dimension, the lower cell, cell (3,3), was judged to yield more intelligible speech. It is thus possible that cells below cell (3,3) would yield higher intelligibility. Consequently, the vertical leg of the L- shaped elemental unit was moved downward by one step to occupy cells (3,2) and (4,2), respectively. Paired comparisons were then performed between the cells on the horizontal and vertical legs ofthe elemental unit 11. The results showed that, on the horizontal axis, cell (3,2) was favored over cell (3,1). This represented a reversal in judgments of relative intelligibility with horizontal movement and, as a consequence, the horizontal leg for elemental unit III was moved one step to the right. The paired-comparisons for the vertical leg of elemental unit II showed that cell (3,2) was favored over cell (4,2). This result indicated a reversal in judgments of relative intelligibility with movement on the vertical axis and the vertical leg of elemental unit III was thus moved one step upward. Paired-comparison judgments among the cells forming elemental unit III show reversals once again on both the horizontal and vertical axes, leading to the choice of elemental unit IV, which is identical in location to elemental unit Il. Pairs of reversals have now been obtained on both the horizontal and vertical axes. These reversals can be used to obtain a first estimate of the cell for maximizing relative intelligibility. Following the recommended estimation procedure for up-down procedures, the average of the peak and valley in the horizontal direction (loudness levels 3 and 1, respectively) yields an estimated optimum loudness level of 2 for the horizontal axis. Similarly, the average of the peak and valley in the vertical direction (loud- ness levels 4 and 2, respectively) yields an estimated optimum loudness level of 3 for the vertical axis (i.e., cell (3,2) is the first estimate of the cell for maximizing relative intelligibility). The two-dimensional up-down procedure described above behaves in much the same way as any other adaptive procedure of the up-down type. As a consequence, the same four rules described earlier for the efficient implementation of up-down procedures apply to this case as well. There are, however, several cautions that need to be noted. Adaptive testing on more than one dimension is much more time consuming than the single dimensional case. Further, the testing time that is needed increases dramatically with each additional dimension. The cost of a poor choice for the initial value is thus much greater in multidimensional adaptive testing (i.e., a considerable number of trials may be wasted with a poor choice of the initial value). It is thus recommended that one of the better prescriptive formulas be used to obtain the initial value. (See Sullivan et al, 1988, for an experimental comparison of several well-known prescriptive procedures.) If the prescriptive formula provides the best fit for a given subject, then only a few adaptive trials will be needed to confirm the optimality of the prescription. If the prescriptive formula is slightly off for a given individual, then a few more adaptive trials will be needed to determine the optimum setting. If the prescriptive formula is completely wrong, then many adaptive trials will be needed to converge on the optimum setting. One of the findings of the Sullivan et al (1988) study is that there may not be a single optimum setting for any given individual (i.e., there may be optimum settings for different acoustic environments). In this case, the optimum settings for commonly encountered acoustic environments should be estimated separately. The hearing aid should then be programmed to switch to the appropriate optimum setting when there is a change in the acoustic environment. This strategy is now possible using modern programmable hearing aids with memory. It is also likely that more advanced hearing aids will adjust automatically to these optimum settings. The self-adapting noise reduction instruments currently on the market represent a step in this direction. In order for advanced hearing aids to function in this way, it is becoming increasingly important for hearing-aid dispensers to identify the optimum settings of a hearing aid as efficiently as possible. 129

A "..ryy.iyv.u WiYa+y~}~f^Wrw':W-': o-.v~u4'i. -' ryi"r~aw..oifiaj4.vyru.. W.rs.rs."ru.rr,u"iY`I~il~.~.aWwYlYa..YtiJaWafu. :.. i173wa : ieludi~~~ Journal of the American Academy of Audiology/Volume 3, Number 2, March 1992 The problem of choosing an appropriate step size for each of the variables to be adjusted is a difficult problem since, at present, we know little about how the relative performance of a hearing aid changes with each variable. Further, Neuman et al (1987) found that the rate at which relative intelligibility changed with the shape of the frequency-gain characteristic varied substantially among subjects. Those subjects showing a sharp peak in intelligibility as a function of relative gain in the low and high frequencies also showed rapid and reliable convergence on the optimum cell. In the absence of detailed knowledge of how the property of interest (e.g., intelligibility, overall sound quality) varies with the parameters being adjusted, it is recommended that the step size correspond to perceptible changes in the stimulus (i.e., changes less than a just noticeable difference by the hearing-impaired subject are not recommended). In some cases, such as the use of the adaptive procedure with programmable hearing aids currently on the market, the step size may be dictated by the settings that are available on the instrument. Even with this constraint, relatively good estimates of the optimum setting are possible, as demonstrated by Kuk and Pape (in press). A more difficult and much more important problem is that of choosing the right variables. In the example given, the two variables chosen for adjustment were relative gain in the low and high frequencies, respectively. This appears to be a reasonable choice, but it is possible that the choice of other variables may have yielded even better results. As discussed by Levitt (1978b), choosing the right variables is perhaps the most important unsolved problem in hearing aid prescription and one that deserves much more attention in research studies. Note that with modern signal processing hearing aids, the choice of variables need not be restricted to traditional considerations such as gain, frequency response, and output limiting, but may also include additional or alternative variables involving advanced methods of signal processing. The method of estimation used in the example given above is a simple extension of that used for up-down testing on a single dimension (i.e., the estimates are obtained, as before, for each variable that is adjusted). This procedure is both simple and efficient provided convergence on the optimum setting occurs at about the same time for each variable. If, for example, several reversals are obtained on the horizontal axis well before there is a pair of reversals on the vertical axis, then only the last pair of reversals on the horizontal axis should be used in order to obtain the first estimate of the optimum setting. Finally, the multivariate adaptive procedure described above is very similar, conceptually, to the simplicial method of adaptive testing in which a simplex (e.g., a triangle in 2- dimensions, a pyramid in 3 dimensions) is used rather than the more convenient L-shaped elemental unit (Box, 1957 ; Levitt, 1978b). The adaptive procedure described above, because of its similarity to the original simplex method, is usually referred to as the modified simplex method (Neuman et al, 1987 ; Kuk and Pape, in press). CONCLUSION T he adaptive procedures described in this paper have a wide range of applications in audiology and are particularly well suited for hearing-aid prescription. For example, in order to prescribe a hearing aid properly it is necessary to know the subject's threshold of hearing, most comfortable listening level, and loudness discomfort level. Each of these variables can be estimated efficiently and reliably using either the simple up-down technique or one of the transformed up-down procedures. A much more difficult problem is that of adjusting the many variables of a modern hearing aid to maximize a desirable property such as speech intelligibility or overall sound quality. A relatively simple version of this problem is that of finding PB-max since it involves optimization of only one variable, the stimulus level. A transformed up-down procedure for estimating PB-max has been derived. Following the same basic approach, the adaptive peak-seeking procedure has been generalized to cover two or more variables. Since the procedure depends on relative rather than absolute judgments of intelligibility (or relative judgments of any other property of interest) it can be used effectively with paired-comparison judgments. Multivariate adaptive methods of hearing aid prescription have been developed using this approach. Preliminary evaluations of these procedures have yielded promising results. All ofthe techniques discussed are based on the simple up-down technique. There are several reasons why this has been done. The simple up-down procedure is both easy to implement, and highly efficient (for its intended applica-