Both-ear Method for the Analysis of Audiometric Data

Abstract

Objective:

Single-ear hearing measurements, such as better-ear, worse-ear or left/right ear, are often used as outcomes in auditory research, yet measurements in the two ears of the same individual are often strongly but not perfectly correlated. We propose a both-ear method using the Generalized Estimating Equation approach for analysis of correlated binary ear data to evaluate determinants of ear-specific outcomes that includes information from both ears of the same individual.

Design:

We first theoretically evaluated bias in odds ratio (OR) estimates based on worse-ear and better-ear hearing outcomes. A simulation study was conducted to compare the finite sample performances of single-ear and both-ear methods in logistic regression models. As an illustrative example, the single-ear and both-ear methods were applied to estimate the association of Dietary Approaches to Stop Hypertension (DASH) adherence scores with hearing threshold elevation among 3,135 women, aged 48–68 years, in the Nurses’ Health Study II.

Results:

Based on statistical theories, the worse-ear and better-ear methods could bias the OR estimates. The simulation results led to the same conclusion. In addition, the simulation results showed that the both-ear method had satisfactory finite sample performance and was more efficient than the single-ear method. In the illustrative example, the confidence intervals of the estimated ORs for the association of DASH scores and hearing threshold elevation using the both-ear method were narrower, indicating greater precision, than for those obtained using the other methods.

Conclusions:

The worse-ear and better-ear methods may lead to biased estimates, and the left/right ear method typically results in less efficient estimates. In certain settings, the both-ear method using the Generalized Estimating Equation approach for analyses of audiometric data may be preferable to the single-ear methods.

INTRODUCTION

Hearing loss is the fourth largest contributor to disability in the world (World Health Organization 2018), thus identifying potential risk factors for hearing loss is a pressing global health issue (Looi et al. 2015). The gold standard for measurement of hearing sensitivity is pure tone audiometry. The pure-tone audiogram provides information in hearing sensitivities across a selected frequency range and is widely used for investigative research and in the clinical setting. Previous research has commonly defined hearing status based on the pure-tone average (PTA), which is the average of hearing threshold measurements over a pre-defined set of audiometric frequencies for an individual ear. For example, in a study estimating the hearing loss prevalence based on PTA for children in the United States (Niskar et al. 1998), hearing loss was categorized according to average sensitivities in the left and right ear, and by the better (lower PTA) and worse (higher PTA) ear.

Although these single-ear methods are valuable in obtaining estimates of hearing loss prevalence and could be useful in the prediction of communication difficulty, a clinically meaningful outcome, it remains a question whether single-ear methods represent the best practice approach for evaluating exposure-hearing associations. Previous studies have used single-ear measurements as the response variable for exploring relationships between risk factors and hearing status (Cruickshanks et al. 2003; Bainbridge et al. 2008; Zhan et al. 2009; Lee et al. 2020). For example, in a study in the National Health and Nutrition Examination Survey (Bainbridge et al. 2008) that evaluated the cross-sectional relation of diabetes and hearing loss, low/mid frequency was defined as PTA _{0.5,1,2 kHz} and high frequency was defined as PTA _{3,4.6.8 kHz}. Hearing loss of mild or greater severity was defined as a PTA>25 dB HL and hearing loss of moderate or greater severity was defined as PTA> 40 dB HL. For each combination of frequency and severity, worse ear was defined as hearing loss in at least one ear, while better ear was defined as hearing loss in both ears. Adjusted odds ratio (OR) estimates were calculated using multivariate logistic regression for the association of diabetes and hearing loss in the worse and the better ear. In a longitudinal study conducted in the Epidemiology of Hearing Loss Study that evaluated risk factors for 5-year incident hearing loss (Cruickshanks et al. 2003), incident hearing loss was defined as PTA_{0.5,1,2,4 kHz} > 25 dB HL in the worse ear at follow-up among those without hearing loss (PTA_{0.5,1,2,4 kHz} ≤25 dB HL) in both ears at baseline.

In this paper, we investigate the validity and efficiency of using single-ear hearing measurement as the response variable in the evaluation of exposure-hearing associations. We also propose a both-ear method and compare the analysis results obtained using this approach with those obtained using traditional single-ear methods (worse-ear, better-ear, and left/right ear methods).

As an illustrative example, we apply both single-ear and both-ear methods to assess the association of the Dietary Approaches to Stop Hypertension (DASH) diet adherence scores and hearing threshold elevation in the Nurses’ Health Study II (NHS II) Conservation of Hearing Study (CHEARS) (Curhan et al. 2020).

THEORETICAL CONSIDERATIONS AND STATISTICAL METHODS

The single-ear method

Let Y_ij denote a binary response variable; Y_ij =1 represents hearing loss or hearing threshold elevation for the jth ear of the ith individual. The single-ear method assumes the following logistic regression model for the single-ear hearing outcome (Cruickshanks et al. 2003; Bainbridge et al. 2008; Curhan et al. 2020).

$$logit\left(E\left[{\tilde{Y}}_i|{\mathit{x}}_{\mathit{i}}\right]\right)=log\left[\frac{P\left({\tilde{Y}}_i=1|1,x_{i1},\cdots x_{iK}\right)}{P\left({\tilde{Y}}_i=0|1,x_{i1},\cdots x_{iK}\right)}\right]={\alpha }_0+\sum _{k=1}^K{\alpha }_k{x}_{ik}$$ (1)

where ${\tilde{Y}}_i$ represents the binary hearing measurement in the worse ear of the ith individual for the worse-ear method and that in the better ear for the better-ear method, E and P denote expectation and probability, respectively, and x_i = (x_i1,…, x_iK) represents the exposure of interest and potential confounders.

The worse-ear and better-ear methods only use the hearing measurement on one selected ear when fitting the logistic regression model. If an individual-level exposure (e.g., the DASH score) influences hearing, the hearing measurements in both ears contain information about the exposure effects. A more reasonable model, which takes advantage of both ears’ data, for charactering the exposure-hearing relationship is

$$logit\left(E\left[Y_{ij}|{\mathit{x}}_{\mathit{i}\mathit{j}}\right]\right)=log\left[\frac{P\left(Y_{ij}=1|1,x_{ij1},\cdots x_{ijK}\right)}{P\left(Y_{ij}=0|1,x_{ij1},\cdots x_{ijK}\right)}\right]={\beta }_0+\sum _{k=1}^K{\beta }_k{x}_{ijk}.$$ (2)

Here Y_ij, j = 1,2, is a clustered response variable with cluster size 2 (left and right ears), x_ij = (x_ij1,…,x_ijK) represents the exposure of interest and potential confounders for the jth ear of the ith individual. Potential effect modifiers (e.g., baseline hearing ability for each ear) can be taken into account in Model (2) by including interactions of the exposure and the ear- and/or individual- level effect modifiers. Next, we show that the estimated regression coefficients from Model (1) may be biased if Model (2) is true.

The hearing measurements in the two ears, Y_i1 for the left ear and Y_i2 for the right ear, can be treated as correlated Bernoulli variables. Denote P(Y_i1 = 1) = pL, P(Y_i2 = 1 p_R, and the correlation coefficient of Y_i1 and Y_i1 as ρ; for notational simplicity, we have omitted subscript i in these parameters. For the worse-ear method,

$$P\left({\tilde{Y}}_i=1\right)=p_L+p_R-\rho \sqrt{p_L\left(1-p_L\right)}\sqrt{p_R\left(1-p_R\right)}-p_L{p}_R.$$ (3)

Similarly, for the better-ear method,

$$P\left({\tilde{Y}}_i=1\right)=\rho \sqrt{p_L\left(1-p_L\right)}\sqrt{p_R\left(1-p_R\right)}+p_L{p}_R.$$ (4)

See the Supplementary Material Appendix A for a detailed derivation. It follows that, if p_L = p_R = p, Formula (3) and (4) convert to (5) and (6), respectively;

$$\text{for}\text{the}\text{worse-ear}\text{method},P\left({\tilde{Y}}_i=1\right)=\left(2-\rho \right)p+\left(\rho -1\right)p^2;$$ (5)

$$\text{for}\text{the}\text{better-ear}\text{method},P\left({\tilde{Y}}_i=1\right)=\rho p+\left(1-\rho \right)p^2.$$ (6)

Formula (3−6) suggest that, in either the worse-ear or the better-ear method, $P\left({\tilde{Y}}_i=1\right),$which is modeled in (1), is typically not equal to the probabilities, p_L or p_R, which is modeled in (2), even in the simple case of p_L = p_R = p. Therefore, the estimated regression coefficients in Model (1) may not converge to the coefficients in Model (2) even if the sample size goes to infinity.

The left or right ear method models $P\left({\tilde{Y}}_i=1\right)=p_L$ or $P\left({\tilde{Y}}_i=1\right)=p_R$. This method is valid as the estimated regression coefficients converge to the coefficients in Model (2); however, it is less efficient than a method using both ears since it uses less data than the both-ear method.

Next, we evaluate the magnitude of the bias in ORs assuming there is only one binary exposure in Models (1) and (2) and, within the exposure group, p_L = p_R = p. Based on Model (2), OR $=\frac{p_e}{1-p_e}/\frac{p_{\stackrel{-}{e}}}{1-p_{\stackrel{-}{e}}}=\frac{p_e\left(1-p_{\stackrel{-}{e}}\right)}{p_{\stackrel{-}{e}}\left(1-p_e\right)}$, where $p_e=\mathrm{P}\left(Y_{ij}=1|X_i=1\right)\mathrm{a}\mathrm{n}\mathrm{d}{p}_{\stackrel{-}{e}}=\mathrm{P}\left(Y_{ij}=1|X_i=0\right),$ representing the prevalence of the hearing outcome being 1 in the jth ear in the exposed and unexposed groups, respectively, j = 1, 2. However, using the worse-ear method, exponential of the regression coefficient in model (1) represents the following OR:

$${OR}_w=\frac{p_{ew}}{1-p_{ew}}/\frac{p_{\stackrel{-}{e}w}}{1-p_{\stackrel{-}{e}w}}=\frac{\left[\left(2-\rho \right)p_e+\left(\rho -1\right)p_e^2\right]\left\{1-\left[\left(2-\rho \right)p_{\stackrel{-}{e}}+\left(\rho -1\right)p_{\stackrel{-}{e}}^2\right]\right\}}{\left[\left(2-\rho \right)p_{\stackrel{-}{e}}+\left(\rho -1\right)p_{\stackrel{-}{e}}^2\right]\left\{1-\left[\left(2-\rho \right)p_e+\left(\rho -1\right)p_e^2\right]\right\}}=\frac{\left[\left(2-\rho \right)+\left(\rho -1\right)\frac{OR\cdot p_{\stackrel{-}{e}}}{OR{p}_{\stackrel{-}{e}}-p_{\stackrel{-}{e}}+1}\right]}{\left\{\frac{OR{\cdot p}_{\stackrel{-}{e}}-p_{\stackrel{-}{e}}+1}{OR\cdot p_{\stackrel{-}{e}}}-\left[\left(2-\rho \right)+\left(\rho -1\right)\frac{OR\cdot p_{\stackrel{-}{e}}}{OR{p}_{\stackrel{-}{e}}-p_{\stackrel{-}{e}}+1}\right]\right\}}\frac{\left\{1-\left[\left(2-\rho \right)p_{\stackrel{-}{e}}+\left(\rho -1\right)p_{\stackrel{-}{e}}^2\right]\right\}}{\left[\left(2-\rho \right)p_{\stackrel{-}{e}}+\left(\rho -1\right)p_{\stackrel{-}{e}}^2\right]},$$ (7)

where $p_{ew}=\mathrm{P}\left({\tilde{Y}}_i=1|X_i=1\right)$ and $p_{\stackrel{-}{e}w}=\mathrm{P}\left({\tilde{Y}}_i=1|X_i=0\right)$, and we have used Formula (5) in the second equality. Note that OR_W = 1 OR = 1; that is, there is no difference between OR and OR_W under the null (i.e., if there is no exposure-hearing association).

For the better-ear method, exponential of the regression coefficient in model (1) corresponds to

$${OR}_B=\frac{p_{eb}}{1-p_{eb}}/\frac{p_{\stackrel{-}{e}b}}{1-p_{\stackrel{-}{e}b}}=\frac{\left[\rho p_e+\left(1-\rho \right)p_e^2\right]\left\{1-\left[\rho p_{\stackrel{-}{e}}+\left(1-\rho \right)p_{\stackrel{-}{e}}^2\right]\right\}}{\left[\rho p_{\stackrel{-}{e}}+\left(1-\rho \right)p_{\stackrel{-}{e}}^2\right]\left\{1-\left[\rho p_e+\left(1-\rho \right)p_e^2\right]\right\}}=\frac{\left[\rho +\left(1-\rho \right)\frac{OR{\cdot p}_{\stackrel{-}{e}}}{OR\cdot p_{\stackrel{-}{e}}-p_{\stackrel{-}{e}}+1}\right]}{\left\{\frac{OR\cdot p_{\stackrel{-}{e}}-p_{\stackrel{-}{e}}+1}{OR\cdot p_{\stackrel{-}{e}}}-\left[\rho +\left(1-\rho \right)\frac{OR\cdot p_{\stackrel{-}{e}}}{OR\cdot p_{\stackrel{-}{e}}-p_{\stackrel{-}{e}}+1}\right]\right\}}\frac{\left\{1-\left[\rho \cdot p_{\stackrel{-}{e}}+\left(1-\rho \right)p_{\stackrel{-}{e}}^2\right]\right\}}{\left[\rho \cdot p_{\stackrel{-}{e}}+\left(1-\rho \right)p_{\stackrel{-}{e}}^2\right]}$$ (8)

where $p_{eb}=\mathrm{P}\left({\tilde{Y}}_i=1|X_i=1\right)$ and $p_{\stackrel{-}{e}b}=\mathrm{P}\left({\tilde{Y}}_i=1|X_i=0\right).$ Similarly, there is no difference between OR and OR_B under the null.

Figure 1 shows numerical results for the relative differences between OR_W (or OR_B) and OR as functions of OR and the between-ear correlation coefficient, ρ, when $p_{\stackrel{-}{e}}$ is 0.5, based on formula (7) and (8). Using the worse-ear method as an example, this relative difference is defined as (OR_W−OR/OR. The relative difference for OR_W ranges from −10% to 11.5% when ρ =0.2. When ρ = 0.8, the difference is close to zero. The between-ear correlation coefficient for the hearing outcome, ≥5 dB HL threshold elevation, in the CHEARS data example is approximately 0.3. The absolute value of the relative difference increases when OR moves away from 1. The relative difference of OR_B is larger than that in the worse-ear method, especially when the ρ is smaller. For example, when ρ = 0.2 and OR = 2.0, the relative difference of the ORs between the better-ear and both-ear methods is approximately 12.5%; when OR = 0.5, this relative difference is close to −13.75%. Supplementary Figure 1 shows the relative differences when $p_{\stackrel{-}{e}}$ = 0.1 where OR_W and OR_B are less different from the OR than those when $p_{\stackrel{-}{e}}$ = 0.50.

Fig. 1.

The relative difference comparing the odds ratio using the worse-ear method (upper panel) or the better-ear method (lower panel) with the both-ear method (${\mathit{P}}_{\stackrel{-}{\mathit{e}}}=0.5$).

The both-ear method

We can use the Generalized Estimating Equation (GEE) method (Liang & Zeger 1986) to make inferences about the parameters in Model (2) based on the correlated both-ear data. GEE is a semiparametric method since we do not need to make distributional assumptions on the response variables. The GEE estimator of β = (β₁, β₂,⋯,β_k) in Model (2) is obtained by solving the following equations:

$$\sum _{i=1}^N{D}_i^{\mathrm{\text{'}}}{V}_i^{-1}\left({\mathit{Y}}_{\mathit{i}}-{\mathit{\mu }}_{\mathit{i}}\right)=0,$$ (10)

where Y_i is the vector of the outcome variable formed by Y_ij, μ_i is the expected value E(Y_i|x_i, and V_i is the working variance-covariance matrix, and D_i = ∂μ_i/∂β. In the GEE method, as long as the mean model, E(Y_i|x_i, is specified correctly, the regression coefficients will be estimated consistently, even if the working variance-covariance matrix is different from cov(Y_i). We can use the sandwich estimator to estimate the variance of the regression coefficient estimates (Fitzmaurice et al. 2011).

Notably, the single-ear method has correlated data if each participant has multiple hearing threshold measures at multiple frequencies or frequency categories, which is typical for audiometric data. When there are threshold measurements at multiple frequencies, for both the both-ear and single-ear methods, data for all the frequencies or frequency categories should be analyzed in the same model and include the exposure-frequency interaction as well as the interactions between frequency and other relevant covariates; GEE method can then be used to estimate the regression coefficients of the regression models, taking into account the within-participant correlation between hearing threshold measurements in both ears and across frequencies.

Using three frequency groups (low, mid, high) as an example, we can create two indicators, x₂ and x₃, for the frequencies; that is, x₂ = 1 for the mid frequencies and 0 otherwise; x₃ = 1 for the high frequencies and 0 otherwise. Let x₁ denote a continuous exposure variable of interest. The component in the right-hand side of Model (2) for the exposure and exposure-frequency interactions can then be written as β₁x₁ + β₂x₁x₂+β₃x₁x₃. It follows that the ORs representing the exposure effect per 1 unit increase in x₁ for low, mid and high frequencies are exp(β₁),exp(β₁ + β₂),exp(β₁ + β₃), respectively. If these ORs for different frequencies are estimated in separate analyses, the correlations between these OR estimates would be ignored, leading to incorrect Type I error in global tests such as the test for whether there is an exposure effect in at least one frequency group. This could result in misleading scientific findings.

SIMULATION STUDY

We conducted a simulation study to compare the four methods considered. Correlated binary responses were generated following methods previously described elsewhere (Touloumis 2016). Similar to the CHEARS study, for each individual (e.g., participant), the binary outcome variable had six dimensions, PTA at low-frequency, mid-frequency and high-frequency for both ears. We assumed: (a) the correlation coefficient, ρ, between different frequencies for the same ear was ρ=0.3; (b) for the same frequencies for different ears, ρ=0.6; and (c) for different frequencies for different ears, ρ=0.2. The GEE approach with an exchangeable working correlation matrix was used in the four methods considered to estimate the regression coefficients of the logistic regression models. R 4.0.3 was used for the simulation study and the R package ‘gee’ was used for the GEE analyses. Four criteria were calculated to compare the bias and efficiency of the estimated exposure coefficients (log(OR)) among the methods:

1. The percent (%) relative bias, defined as 100*(the mean of the estimated over simulation replicates – the true coefficient)/the true coefficient;

2. The empirical standard deviation of the point estimates (a smaller value represents a more efficient estimate);

3. The average of the estimated standard errors of the estimated coefficients;

4. The 95% confidence interval coverage rate (a converge rate closer to 95% indicating a better interval estimates).

We considered the two simulation scenarios, described below.

In the first simulation scenario, we included the following terms in logistic regression model: x₁, a continuous variable for the exposure, participant-specific; x₂ and x₃, the dummy variables for mid- and high-frequency, respectively, participant-specific; interactions x₁ × x₂ and x₁ × x₃; x₄, the baseline PTA measurements, ear-specific. The true coefficients of these six terms were all set as log(1.5). The number of simulation replicates was 1000, and the sample size (i.e., the number of 6-dimention clusters) was 1000 for each simulation replicate. The simulation results are shown in Table 1 (for 20% event rate) and Supplementary Table 1 (for 50% event rate). In this simulation scenario, we also conducted an additional simulation study for the 20% event rate using the unstructured working correlation matrix in the GEE analyses; the simulation results are shown in Supplementary Table 2.

TABLE 1:

Comparison of Alternative Analytic Methods for the Analysis of Audiometric Data: Simulation Study Scenario 1^a

	Both-ear Methodb				Worse-ear Methodc				Better-ear Methodd				Left-ear Methode
	Relative biasf (%)	Empirical standard deviationg	Estimated standard errorh	Converge ratei (%)	Relative bias (%)	Empirical standard deviation	Estimated standard error	Converge rate (%)	Relative bias (%)	Empirical standard deviation	Estimated standard error	Converge rate (%)	Relative bias (%)	Empirical standard deviation	Estimated standard error	Converge rate (%)
x1	−0.34	0.07	0.07	0.94	4.62	0.08	0.08	0.95	48.91	0.17	0.18	0.76	0.12	0.09	0.09	0.95
x2	−0.73	0.08	0.08	0.95	8.61	0.09	0.09	0.93	47.78	0.23	0.22	0.89	0.37	0.12	0.12	0.96
x3	−0.32	0.08	0.08	0.96	9.18	0.09	0.09	0.94	48.90	0.23	0.22	0.88	−0.11	0.12	0.12	0.94
x1×x2	0.78	0.08	0.08	0.95	14.27	0.10	0.10	0.91	25.90	0.19	0.20	0.90	0.51	0.12	0.12	0.95
x1×x2	0.40	0.08	0.08	0.96	14.48	0.10	0.10	0.91	24.27	0.19	0.19	0.92	0.83	0.12	0.12	0.95
x4	0.30	0.04	0.04	0.95	−1.51	0.05	0.05	0.95	5.16	0.08	0.09	0.92	0.22	0.05	0.05	0.96

^aSimulation Study Scenario with a continuous exposure, a 20% event rate and a sample size of 1000, based on 1000 simulation replicates; using exchangeable working covariance-covariance matrix.

^bBoth-ear method: logistic regression with hearing data of both ears at three frequency categories as correlated outcome.

^cWorse-ear method: logistic regression with hearing data of the worse ear at three frequency categories as the outcome.

^dBetter-ear method: logistic regression with hearing data of the better ear at three frequency categories as the outcome.

^eLeft ear method: logistic regression with hearing data of the left ear at three frequency categories as the outcome; the generalized estimating equation approach was used in all the methods for estimation.

^fRelative Bias is 100 × (the mean of the estimated log (OR) over simulation replicates – the true log (OR))/the true log (OR), where OR stands for odds ratio.

^gEmpirical standard deviation (SD) is the empirical SD of the log(OR) estimates from simulation replicates.

^hEstimated standard error (SE) is the average of the sandwich SE over the simulation replications.

ⁱCoverage Rate is the 95% confidence interval coverage rate based on the sandwich SE.

x₁: person-level continuous exposure; x₂: dummy variable for mid-frequency; x₃: dummy variable for high-frequency; x₄ continuous ear-level baseline measurement.

In the second simulation scenario, instead of the continuous exposure evaluated in the first scenario, we examined a four-level categorical exposure. For this categorical exposure, we created the indicator variables x_1,1, x_1,2, and x_1,3. We therefore used the interaction terms: x_1,1 × x₂, x_1,2 × x₂, x_1,3 × x₂, x_1,1 × x₃, x_1,2 × x₃ and x_1,3 × x₃. We set the regression coefficients of these terms similar to the estimated regression coefficients in the both-ear analysis for the CHEARS data (Burton et al., 2006), and the sample sizes in each simulation replicate were 1000 and 5000. The corresponding results are shown in Tables 2 (for 20% event rate and sample size 1000), Supplementary Table 3 (for 50% event rate and sample size 1000), Tables 3 (for 20% event rate and sample size 5000) and Supplementary Table 4 (for 50% event rate and sample size 5000).

TABLE 2:

Comparison of Alternative Analytic Methods for the Analysis of Audiometric Data: Simulation Study Scenario 2 ^a

	Both-ear methodb				Worse-ear methodc
	Relative biasd (%)	Empirical standard deviatione	Estimated standard errorf	Converge rateg (%)	Relative bias (%)	Empirical standard Deviation	Estimated standard error	Converge rate (%)
x1,1	−3.51	0.24	0.25	0.95	−13.90	0.25	0.27	0.94
x1,2	2.51	0.24	0.25	0.95	−1.63	0.27	0.29	0.94
x1,3	5.62	0.23	0.24	0.95	−2.91	0.24	0.25	0.94
x2	1.76	0.20	0.21	0.95	3.93	0.21	0.22	0.93
x3	0.90	0.21	0.22	0.94	8.24	0.23	0.24	0.90
x1,1×x2	−2.04	0.30	0.31	0.94	−11.04	0.32	0.33	0.94
x1,2×x2	1.52	0.27	0.28	0.95	9.28	0.31	0.33	0.94
x1,3×x2	38.73	0.28	0.29	0.94	−1.09	0.30	0.31	0.95
x1,1×x3	23.29	0.30	0.32	0.94	−35.35	0.33	0.34	0.94
x1,2×x3	2.85	0.29	0.31	0.94	25.40	0.34	0.37	0.93
x1,3×x3	−7.50	0.26	0.28	0.94	127.11	0.30	0.31	0.93
x4	−6.67	0.04	0.04	0.95	−17.82	0.04	42.00	0.96

^aSimulation Study Scenario with a 4-level categorical exposure, a 20% event rate and a sample size of 1000, based on 1000 simulation replicates; using exchangeable working covariance-covariance matrix.

^bBoth-ear method: logistic regression with hearing data of both ears at three frequency categories as correlated outcome.

^cWorse-ear method: logistic regression with hearing data of the worse ear at three frequency categories as the outcome.

^dRelative Bias is 100 × (the mean of the estimated log (OR) over simulation replicates – the true log (OR))/the true log (OR), where OR stands for odds ratio.

^eEmpirical standard deviation (SD) is the empirical SD of the log(OR) estimates from simulation replicates.

^fEstimated standard error (SE) is the average of the sandwich SE over the simulation replications.

^gCoverage Rate is the 95% confidence interval coverage rate based on the sandwich SE.

x_1,1, x_1,2, and x_1,3: indicators for a categorical exposure; x₂: dummy variable for mid-frequency; x₃: dummy variable for high-frequency; x₄ continuous ear-level baseline measurement.

The better-ear and left/right-ear methods have low convergence rates and thus the results are unavailable.

TABLE 3:

Comparison of Alternative Analytic Methods for the Analysis of Audiometric Data: Simulation Study Scenario 2^a

	Both-ear Methodb				Worse-ear Methodc				Better-ear Methodd				Left-ear Methode
	Relative biasf (%)	Empirical standard deviationg	Estimated standard errorh	Converge ratei (%)	Relative bias (%)	Empirical standard deviation	Estimated standard error	Converge rate (%)	Relative bias (%)	Empirical standard deviation	Estimated standard error	Converge rate (%)	Relative bias (%)	Empirical standard deviation	Estimated standard error	Converge rate (%)
x1,1	−2.06	0.11	0.11	0.95	−12.48	0.11	0.11	0.94	−143.41	0.34	0.35	0.84	−1.18	0.15	0.15	0.94
x1,2	0.36	0.11	0.11	0.94	−1.00	0.12	0.12	0.95	30.64	0.35	0.35	0.96	1.45	0.15	0.15	0.95
x1,3	−0.87	0.10	0.11	0.95	6.45	0.11	0.11	0.94	−67.34	0.32	0.32	0.91	0.13	0.15	0.14	0.96
x2	0.28	−0.09	0.09	0.94	2.86	0.09	0.10	0.93	37.71	0.26	0.26	0.71	0.68	0.13	0.13	0.96
x3	−0.06	0.09	0.09	0.94	7.69	0.10	0.10	0.75	42.97	0.25	0.25	0.24	0.11	0.13	0.13	0.96
x1,1×x2	−1.86	0.13	0.14	0.95	−10.85	0.14	0.14	0.94	−129.65	0.38	0.40	0.87	1.86	0.18	0.19	0.95
x1,2×x2	5.32	0.12	0.12	0.94	4.38	0.13	0.14	0.94	64.38	0.38	0.38	0.95	−9.85	0.17	0.18	0.95
x1,3×x2	0.76	0.12	0.13	0.94	−13.68	0.13	0.13	0.94	−118.55	0.36	0.37	0.92	4.18	0.17	0.17	0.95
x1,1×x3	10.00	0.13	0.14	0.94	−39.86	0.14	0.15	0.94	269.03	0.37	0.38	0.94	34.17	0.18	0.19	0.94
x1,2×x3	0.35	0.13	0.13	0.95	34.94	0.15	0.15	0.94	1159.58	0.39	0.38	0.96	0.57	0.18	0.18	0.95
x1,3×x3	1.48	0.12	0.12	0.94	98.149	0.13	0.14	0.93	−111.93	0.34	0.35	0.93	−9.17	0.17	0.17	0.95
x4	−0.43	0.02	0.02	0.96	−4.05	0.02	0.02	0.95	38.25	0.04	0.03	0.96	8.94	0.02	0.02	0.95

^aSimulation Study Scenario with a 4-level categorical exposure, a 20% event rate and a sample size of 5000, based on 1000 simulation replicates; using exchangeable working covariance-covariance matrix.

^bBoth-ear method: logistic regression with hearing data of both ears at three frequency categories as correlated outcome.

^cWorse-ear method: logistic regression with hearing data of the worse ear at three frequency categories as the outcome.

^dBetter-ear method: logistic regression with hearing data of the better ear at three frequency categories as the outcome.

^eLeft ear method: logistic regression with hearing data of the left ear at three frequency categories as the outcome; the generalized estimating equation approach was used in all the methods for estimation.

^fRelative Bias is 100 × (the mean of the estimated log (OR) over simulation replicates – the true log (OR))/the true log (OR) ), where OR stands for odds ratio.

^gEmpirical standard deviation (SD) is the empirical SD of the log(OR) estimates from simulation replicates.

^hEstimated standard error (SE) is the average of the sandwich SE over the simulation replications.

ⁱCoverage Rate is the 95% confidence interval coverage rate based on the sandwich SE.

x_1,1, x_1,2, and x_1,3: indicators for a categorical exposure; x₂: dummy variable for mid-frequency; x₃: dummy variable for high-frequency; x₄ continuous ear-level baseline measurement.

The both-ear method outperformed the single-ear methods in all of the simulation scenarios. In the first simulation scenario, for a 20% event rate (Table 1), the left/right ear method led to estimates with less than 1% relative biases, similar to the both-ear method. However, there was an up to 50% larger empirical standard error than that observed using the both-ear method. Both the worse-ear and better-ear method estimators were more biased and had larger empirical standard deviations of the point estimates; the absolute values of the relative biases ranged from 5% to 49% for the better-ear estimator, and from 2% to 14% for the worse-ear estimators. Similar findings were observed for the 50% event rate (Supplementary Table 1); for the worse-ear method estimator, a higher relative bias that ranged from 0.1% to 29%, was observed. Similar findings were also observed when using the unstructured working correlation matrix in the GEEs (Supplementary Table 2).

In the second set of simulation scenarios, for a 20% event rate scenario with a sample size of 1000 (Table 2), the standard logistic regressions in the better-ear and left/right ear methods did not converge in more than 20% of simulation replicates; thus, the simulation results for these two methods are not available. The both-ear method estimator had less finite sample bias and less empirical standard deviation of the point estimates than the worse-ear method. When the sample size was increased to 5000 (Table 3) we observed similar patterns as those observed in the first simulation scenario. For the 50% event rate scenarios, the patterns of the performance of the four methods were similar to those observed in the first simulation scenario (Supplementary Tables 3 and 4).

CHEARS DATA ANALYSIS

To illustrate the implementation of our proposed methods, we provide an illustrative example based on data obtained in the CHEARS prospective study of dietary patterns and hearing threshold elevation (Curhan et al. 2020). Specifically, our example explores data obtained from an investigation of the dietary adherence scores for the DASH diet and risk of 3-year hearing threshold elevation.

CHEARS examines risk factors for hearing loss among participants in several large ongoing cohorts, including the NHS II, an ongoing cohort study of 116,430 female registered nurses in the United States, aged 25–42 years at enrollment in 1989. In a sub-cohort of these NHS II participants, the CHEARS Audiology Assessment Arm (AAA), longitudinal changes in pure-tone air and bone conduction audiometric hearing thresholds were assessed. The methods for the CHEARS AAA are described elsewhere (Curhan et al. 2020, 2021). Briefly, pure-tone air conduction hearing thresholds were assessed at 19 geographically diverse testing sites, conducted by licensed audiologists using standardized protocols and equipment calibrated data to meet American National Standards Institute standards. A priori, participants who reported excellent or good hearing and no history of otologic disease were invited to participate to examine early threshold changes. In total, 3,749 participants completed baseline testing and 3,135 participants (84%) completed 3-year follow-up testing.

The outcome was a binary variable with elevated hearing threshold, defined as at least 5 dB HL elevation in PTA hearing measurements from baseline to year 3. For each participant, the outcome variable had six dimensions, PTA at low-frequency (0.5,1,2 kHz), mid-frequency (3,4 kHz) and high-frequency (6,8 kHz) for both ears. The exposure of interest was the DASH score in quartiles. We adjusted for baseline PTA measurements at low-frequency (0.5,1,2 kHz), mid-frequency (3,4 kHz) and high-frequency (6,8 kHz) for both ears, and the following potential confounders: age (continuous), race (black/white/multi/other or unknown), body mass index (<25kg/m²/25–29/30–34/35–39/40+), smoking status (never/past/current), history of tinnitus (yes/no), cumulative average energy intake (continuous), noise exposure (very loud occupational or leisure-time noise exposure >=3 hours/week during any decade; yes/no). Exposure-frequency interactions were included in all the models. A GEE with the exchangeable covariance matrix was used for all the methods. In total, there were 6270 ears in the analysis from 3135 participants. Based on the worse-ear method, there were a total of 4,627 hearing threshold elevation events across the low-, mid- and high-frequency PTA categories. There were 1,858 events based on the better-ear methods, 3,228 events based on left-ear method, and 3,257 events based on right-ear method. Supplementary Table 4 presents the baseline characteristics of the participants.

Table 4 presents the estimated ORs and confidence intervals representing the DASH-hearing threshold elevation association which were obtained based on the regression coefficient estimates of DASH and interactions between DASH and frequency level. Notably, the confidence intervals of the estimates using both-ear methods were substantially narrower than those obtained using the other methods. For example, for the mid- and high-frequency PTAs, the widths of the confidence intervals of the estimated coefficients for quartile 4 DASH score from the worse-ear method were 21% and 27% wider, respectively, than those obtained with the both-ear method. For the better-ear method, the width of the confidence intervals obtained were 43% and 68% wider than those obtained with the both-ear method. The point estimates were not substantially different between the both-ear and worse/better-ear methods. The better-ear method typically led to estimates with the widest confidence intervals.

TABLE 4:

The Association between Dietary Approaches to Stop Hypertension (DASH) Score and 3-year Audiometric Hearing Threshold Elevation in the Conservation of Hearing Study (CHEARS) Audiology Assessment Arm, 2012–2018

Dash score (quartile)	Both Ear		Worse Ear		Better Ear		Left Ear		Right Ear
	ORa	95% CI	ORa	95% CI	ORa	95% CI	ORa	95% CI	ORa	95% CI
Low-frequency PTA
Quartile 1	1.00	Reference	1.00	Reference	1.00	Reference	1.00	Reference	1.00	Reference
Quartile 2	1.06	0.86,1.29	1.08	0.87,1.34	1.04	0.72,1.50	1.09	0.85,1.41	1.03	0.80,1.32
Quartile 3	0.97	0.79,1.20	0.98	0.79,1.23	1.00	0.69,1.44	1.02	0.79,1.33	0.94	0.73,1.21
Quartile 4	0.86	0.69,1.06	0.88	0.70,1.11	0.84	0.57,1.23	0.92	0.70,1.21	0.82	0.63,1.07
Mid-frequency PTA
Quartile 1	1.00	Reference	1.00	Reference	1.00	Reference	1.00	Reference	1.00	Reference
Quartile 2	1.02	0.86,1.21	1.00	0.82,1.22	1.05	0.83,1.33	1.13	0.92,1.38	0.91	0.74,1.12
Quartile 3	0.91	0.76,1.08	0.87	0.71,1.07	0.95	0.75,1.22	0.95	0.77,1.17	0.87	0.71,1.08
Quartile 4	0.78	0.65,0.93	0.79	0.64,0.98	0.71	0.55,0.92	0.82	0.66,1.01	0.75	0.60,0.93
High-frequency PTA
Quartile 1	1.00	Reference	1.00	Reference	1.00	Reference	1.00	Reference	1.00	Reference
Quartile 2	0.99	0.84,1.16	0.99	0.80,1.22	1.02	0.82,1.27	1.05	0.86,1.28	0.94	0.77,1.15
Quartile 3	0.91	0.77,1.07	0.85	0.69,1.05	0.99	0.80,1.24	0.86	0.70,1.05	0.98	0.80,1.20
Quartile 4	0.88	0.74,1.04	0.86	0.69,1.07	0.91	0.72,1.15	0.79	0.64,0.97	1.01	0.82,1.24

^aOdds ratio based on multivariate logistic regression models adjusted for age (continuous), race (black/white/multi/other or unknown), body mass index(<25kg/m2/25–29/30–34/35–39/40+), smoking status (never/past/current), history of tinnitus (yes/no), cumulative average energy intake (continuous), noise exposure (very loud occupational or leisure-time noise exposure >=3 hours/week during any decade; yes/no), and the ear and frequency-level baseline pure tone average (PTA) measurements at low-frequency PTA (0.5,1,2 kHz), mid-frequency PTA (3,4 kHz) and high-frequency PTA (6,8 kHz) for both ears. Exposure-frequency interactions were also included in all the models.

DISCUSSION

In this paper, we propose a both-ear method for analysis of correlated ear data to evaluate the associations of exposures with hearing outcomes. The traditional single-ear methods, such as the worse-ear and better-ear methods, may lead to biased and inefficient association estimates. The proposed both-ear method, which uses all of the information in the data from both ears, while taking into account within-participant correlation in the analysis, leads to asymptotically unbiased association estimates.

We evaluated the existing single-ear methodologic approaches for analyzing audiometric data, the worse-ear, better-ear and left/right ear methods, and compared these approaches with the proposed both-ear method. First, based on theoretic statistical inferences, we revealed that the bias in the OR estimates based on the worse-ear or better-ear method increased when the within-participant between-ear correlation coefficient for the hearing measurements is smaller and the true effect is stronger. The results from the both-ear method can be similar to the single ear methods when the within-participant between-ear correlation coefficient is large, and the true effect is small. Second, the results of our simulation study suggest that the both-ear method outperforms the three conventional single-ear analytic methods in finite sample bias and in efficiency. The left/right-ear method has better performance with respect to bias, but has lower efficiency than the both-ear method. In the analysis results of the CHEARS data, the robust standard errors using both-ear method are smaller than those using the other methods. This is consistent with the findings in the simulation study.

The GEE approach has been used to estimate population-average effects while taking into account the dependency due to multiple measures over time in longitudinal audiometric data (Rigters et al., 2018). The generalized linear mixed model has been used to estimate subject-specific effects in audiometric data when the hearing threshold is treated as a continuous outcome (Gebregziabher et al., 2018; Kiely et. al., 2012). In this paper we use the GEE method to take into account the within-participants between-ear and between-frequency correlations. The GEE method does not require a distributional assumption and the GEE-estimators are consistent, even if the covariance of response variables is mis-specified. It is noteworthy that even though this paper is focused on logistic regression, the strategy of using both-ear data, with the GEE for making inferences, would apply to other commonly used modeling approaches, including linear regression for continuous outcomes and polytomous logistic regression for more than two-level categorical outcomes.

In the situation where a threshold value is missing, this is because a threshold could not be confidently determined by the audiologist at that frequency. In the CHEARS data, this occurred in a very small number (0.5%) of individuals. Typically, this would be for a single threshold and in only one ear and missing completely at random is a reasonable assumption. Therefore, for those participants with incomplete data we suggest including their partial data that are available in the analysis. In the situations where the missingness is likely to be missing at random, the inverse probability weighting method could be used to handle missingness in the GEE method (Robins et al., 1995).

Although the single-ear methods can be useful in predicting the clinically meaningful outcomes such as communication difficulty, for research investigations that aim to evaluate factors that may influence changes in hearing thresholds, the both-ear methods may be preferable due to the gains in accuracy and efficiency afforded by using information from both ears. The both-ear method could thereby improve the ability to detect meaningful exposure-hearing threshold associations, identify potentially modifiable risk factors for hearing loss, and detect threshold changes at earlier stage when it may be more possible to prevent progression. For example, the both-ear method would be useful in studies for which there is no a priori hypothesis that the exposure(s) of interest differentially influence threshold change in one ear versus the other, such as systemic conditions, dietary intake, physical activity, obesity, or exposure to air pollution. Moreover, in the setting of an a priori hypothesis that the exposure of interest may differentially influence threshold change in one ear versus the other, an interaction between the exposure (e.g., how often a gun is discharged for recreational or occupational reasons) and ear side (e.g., the ear side of discharging a gun), as well as exposure-frequency-ear side interaction if applicable, may be included in the model in addition to the exposure and potential confounders. In this setting, the proposed both-ear analytic approach based on GEE would still be useful for obtaining reliable and precise estimates of the associations observed for each ear.

CONCLUSION

The worse-ear and better-ear methods may lead to biased estimates, and the left/right ear method typically results in less efficient estimates. In certain settings, the both-ear method using the GEE approach for analyses of audiometric data may be preferable to the single-ear methods. Particularly when the research studies are designed to evaluate exposure-hearing threshold relationships, the proposed both-ear analytic approach could be used as the standard method for obtaining reliable and precise estimates of exposure-hearing loss associations.

Supplementary Material

Supplemental Data File (.doc, .tif, pdf, etc.)_2

Supplementary Fig. 1. The relative difference (RD) comparing the odds ratio using the worse-ear method (upper panel) or the better-ear method (lower panel) with the both-ear method (${\mathit{P}}_{\stackrel{-}{\mathit{e}}}=0.1$).