The top panel illustrates the air-conduction thresholds and bone- conduction thresholds for a typical normal ear ͑ solid lines ͒ and impaired ear ͑ dashed lines ͒ . The bottom panel illustrates the corresponding air–bone gaps in decibels. 

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Context 1

... approach is that the errors associated with measuring bone-conduction thresholds are probably higher than those associated with measuring air- conduction thresholds ͑ see sec. III ͒ . That is, the gold standard may be flawed. The likely outcome of this type of error would be an observed reduction in the test efficacy of any predictor, but the comparison of a number of acoustic predictors should remain valid. Another difference between the assessment of cochlear and conductive hearing loss is that cochlear hearing loss is defined at each test frequency typically using as a predictor some EOAE response at the same test frequency. In contrast, conductive hearing loss is often defined as present or absent based upon the air–bone gap at any or all frequencies, but using as a predictor only the 226-Hz tympanogram. We pro- pose a number of different gold standards for conductive hearing loss, either based upon a criterion value of the air– bone gap at a single test frequency ͑ 0.5 or 2 kHz ͒ , or based upon a criterion value of the maximum air–bone gap across all test frequencies. We have used both tympanometric and YR responses as acoustic predictors. In the latter case, the analyses are inherently multivariate, resulting in the combination of responses across frequencies. The test group consisted of a typical clinical population of children with symptoms of otitis media in addition to a baseline population of normal, asymptomatic children. The study was based on data from a subject group ranging in age from 2 to 10 yr ͑ 92 subjects, 174 ears ͒ . We had respectively 10, 100, and 54 ears in our final group of 164 grouped into ranges of 2–3 years old, 4–6 years old, and 7–10 years old. Subjects were drawn from a group of children who were seen through the Boys Town National Research Hospital ENT clinic or whose parents responded to verbal solicitation for participation in a research project. In order to be included in the study the following measurements had to be successfully obtained for each child for at least one ear: air-conduction thresholds for at least 0.5 and 2 kHz, bone-conduction thresholds for at least 0.5 kHz for the same ear, a tympanogram, and an admittance-reflectance ͑ YR ͒ measurement. Audiometric thresholds, acoustic immittance, and admittance-reflectance ͑ YR ͒ measurements were all obtained during the same test session. Otologic data were not available in this study. Audiometric threshold evaluation was performed by a clinical audiologist using either conditioned play audiometry ͑ CPA ͒ , in 104 out of 164 cases, or conven- tional audiometric techniques ͑ CONV ͒ , in 58 out of 164 cases, depending on the developmental level of each subject. Pure-tone air-conduction ͑ AC ͒ and bone-conduction ͑ BC ͒ thresholds were obtained at 0.5 kHz, 2 kHz and, if possible, at 1 and 4 kHz, with a standard diagnostic audiometer Grason-Stadler, GSI-16 linked to insert or dynamic earphones ͑ Etymotic ER-3A or Telephonic TDH-50P ͒ . Bone-conduction thresholds were measured using masked or unmasked condition. Our subjects were drawn from the population seen in an audiology clinic where an air–bone gap equal to or less than 10 dB is considered within normal limits. When the unmasked air–bone gap exceeded 10 dB in a given subject, effective masking levels of narrow- band noise were applied to the nontest ear during bone- conduction testing to ensure against participation of that ear in the determination of threshold. Potential subjects with unmasked air–bone gaps greater than 10 dB who could not be tested successfully using masking were not included in the study. The ratio of masked versus unmasked bone conduction varied from one frequency to another, i.e., at 500 Hz we obtained 53 unmasked and 108 masked BC thresholds, while at 2000 Hz we had 90 unmasked and 52 masked BC thresholds. These data were acquired for the subject within a double-walled, sound-attenuating Industrial Acoustic Com- pany chamber. For subjects in discomfort, often related to middle-ear symptoms, it was sometimes not possible to obtain threshold data at all frequencies. Tympanometry was performed in the audiology clinic using a standard 226-Hz immittance device ͑ Grason-Stadler, GSI-33 Middle Ear Analyzer ͒ . For numerical purposes, 1 the relative gradient rather than tympanometric width was used to record tympanometric gradient ͑ Gr ͒ , defined as follows. The fractional height of a single-peaked tympanogram is the vertical distance between the peak and the point where the compliance pattern is 100 daPA wide. The total height of the compliance pattern is the vertical distance between the peak and the tail as measured from the starting pressure point. The relative gradient is the dimensionless ratio of the fractional height to the total height, so that a flat tympanogram has a relative gradient equal to zero. Examples of audiometric data obtained from a normal ear and an impaired ear classified as having a conductive hearing loss are shown in Fig. 1. The top panel shows the air-conduction ͑ AC ͒ thresholds and the bone-conduction ͑ BC ͒ thresholds, while the bottom panel shows the air–bone gap, calculated as the difference ͑ in decibels ͒ between the AC and BC thresholds. The BC thresholds, indicative of cochlear sensitivity, are similar in both subjects, while the AC thresholds, indicative of transmission through the middle-ear pathway to the cochlea, increased in the impaired ear. The maximum air–bone gap in the impaired ear occurred at 1 kHz, and had a value of 40 dB relative to the normal air– bone gap of 0 dB for an average healthy ear. The only ex- ception in the normal ear occurred at 250 Hz, where a 5-dB air–bone gap was observed. Even though such a small air– bone gap would not be considered clinically significant, all responses at 250 Hz were not included in further analyses, because of an increased sensitivity to error due to inadequate earphone fit at low frequencies. The methodology for measuring input admittance and energy reflectance has been described previously in detail Keefe et al. , 1992 . Briefly, the input signals were produced by a sound source and the responses were measured by a pair of microphones, with all transducers housed in an Etymotic ER-10C probe assembly. A calibration procedure was conducted using a broadband chirp stimulus ͑ 250–10 700 Hz ͒ in five calibration tubes of different lengths, each closed at its opposite end. This calibration provided the Thevenin pressure and Thevenin impedance associated with the probe and measurement system by fitting the measured pressure responses in the tube set to the modeled impedance functions. The model was based upon a cylindrical-tube geometry, including an ideal closed end. After calibration on each day, the probe was inserted into the ear to be tested, and data were acquired using the same broadband chirp stimulus. Given the Thevenin parameters from calibration, the acoustic admittance at the tip of the probe assembly was calculated using the measured pressure response. The energy reflectance was calculated in terms of the measured admittance using the cross-sectional area of the calibration tubes, which all had the same area. This is a modification of the earlier methodology, which used the cross-sectional area inferred from the measurement of the acoustic resistance ͑ the real part of the inverse of the acoustic admittance ͒ . Two sets of calibration tubes were available based on the size of a foam ear tip ͑ ‘‘child’’ or ‘‘adult’’ ͒ most suitable for each subject. This ear tip was then used in the calibration procedure with one or the other of the tube sets approximating the diameter of the ear into which the selected ear tip would comfortably fit. The YR test is most accurate when the ear canal area is within 20% of the actual calibration tube area ͑ Keefe et al. , 1993 ͒ . The complex acoustic admittance is expressed in terms of its real part, the conductance ͑ G ͒ , and its imaginary part, the susceptance ͑ B ͒ ...

Context 2

... the characteristic impedance Z c of a cylindrical tube of area S , similar to the ear-canal area, is Z c ϭ ␳ c / S . The energy reflectance represents the ratio of the reflected to in- cident energy. Its value is relatively insensitive to differences in probe location ͑ assuming negligible losses in the ear canal and the adequacy of a cylindrical-tube model for transmission in the ear canal ͒ . The conductance, equivalent volume, and reflectance are the three YR variables chosen to represent the acoustic response of each ear. Because any pair of these variables can be used to calculate the remaining variable, there is a redun- dancy implicit in this representation. However, some pairs of variables may be better predictors than other pairs. This re- dundancy was retained to test which combination of YR variables has the greatest predictive efficacy. All YR variables were initially stored as 1/12-octave averaged responses, but further averaging over each third octave or octave was applied to reduce the number of YR variables that were input to subsequent statistical analyses. Figure 2 presents examples of reflectance data for the same pair of normal and impaired ears as in Fig. 1. From top to bottom, the conductance ͑ G ͒ , energy reflectance ( ͉ R ͉ 2 ), and equivalent volume ( V ) are plotted as a function of frequency. The energy reflectance is much higher in the mid- frequency region ͑ 1–4 kHz ͒ for the impaired ear than for the normal ear, and the equivalent volume at frequencies below 2 kHz is smaller for the impaired ear than for the normal ear. The resonance just above 4 kHz, which appears in the conductance and equivalent volume as well as the reflectance, has a higher quality factor for the impaired ear. Thus, the YR responses in this pair of ears are clearly discernible. The auditory status of subjects included in the study ranged from normal to moderate conductive hearing loss. Based on the audiological evaluation, all subjects with sen- sorineural and mixed losses, or with perforated tympanic membranes and patent tubes, were excluded from subsequent analyses. In order to assure the hermetic seal of the probe in the ear canal, the YR data sets were evaluated for non-negative equivalent volume measurements, after averaging the response over frequencies from 250 to 1000 Hz. This criterion is based on the fact that the ears of subjects in this age range are compliance dominated at low frequencies ͑ implicit in the results of Keefe et al. , 1993 ͒ . Thus, if the equivalent volume were negative at low frequencies, this would indicate a leak in the placement of the probe assembly within the ear canal. Such a low-frequency leak forms a low-impedance acoustic pathway in parallel with the ear-canal pathway, and its mass- like inertance at low frequencies dominates the calculated equivalent volume. This inclusion criterion identified 3 of 164 YR responses as invalid, corresponding to a data rejection rate of less than 2%. Upon excluding these 3 cases, there were a total of 161 ears used in the main set of analyses. Clinical decision theory is useful in comparing the efficacies of alternative diagnostic tests that discriminate diseased from normal cases ͑ Swets and Pickett, 1982; Swets, 1988 ͒ . In past hearing-research studies using ROC analyses, clinical decision theory has been limited to discrimination between two states or outcomes based on the use of a single input test result. Without loss of generality, we can assume that the higher value of the test result indicates the absence of an abnormality. The true status of a case can be established based on a so-called ‘‘gold standard.’’ For test results which are binary, the accuracy is defined by a pair of parameters— sensitivity or hit rate, and specificity or correct rejection rate. Sensitivity is defined as the probability that the test result is positive given that the ear is impaired, and specificity is defined as the probability that the test result is negative given that the ear is normal. There are two other probabilities in this 2-by-2 decision matrix: the false-alarm rate, which is the probability that the test result is positive given that the ear is normal, and the miss rate, which is the probability that the test result is negative given that the ear is impaired. The sensitivity and specificity values are subject to variation on two independent dimensions: ͑ 1 ͒ the test’s ca- pacity to discriminate an impaired from a normal state, and ͑ 2 ͒ the decision criterion, or gold standard, that is adopted for declaring a test result to be positive ͑ Swets and Pickett, 1982 ͒ . In typical practice, the decision criterion is varied over the full range of values with a pair of sensitivity and specificity values measured for each choice of decision criterion. The estimate of the area under the ROC curve quan- tifies the performance of any particular diagnostic test in a bias-free manner, and can range from 0.5 for a test with no diagnostic ability up to a value of 1.0 for a test with perfect diagnostic ability. The area under the ROC curve can be calculated non- parametrically without having to generate the ROC curve ͑ Bamber, 1975; Hanley and McNeil, 1982 ͒ with the area estimate slightly below that of a maximum likelihood estimate. We used the Wilcoxon–Mann–Whitney ͑ WMW ͒ statistic to estimate area under ROC curves, which employs a trapezoidal rule for integrating the area under the curve. It is biased below the true theoretical value of the ROC area ͑ Song, 1997 ͒ . 2 Alternatively, it is sometimes useful to adopt a particular decision criterion, particularly when there is a priori knowledge of how the test might be used. It may be useful to investigate the performance of various diagnostic tests relative to a fixed, high value of sensitivity if the test is designed to correctly identify most of the impaired ears as impaired. The specificity of each test would be calculated at this fixed sensitivity, and the test with the highest specificity would be the ‘‘best’’ at this reference sensitivity. In comparing test performance for prediction of a conductive hearing loss, we examined both the area under the ROC ...