Analytical methods for correlated data arising from multicenter hearing studies

Abstract

In epidemiological hearing studies, estimating the association between exposures and hearing loss using audiometrically-assessed hearing measurements is challenging due to the complex correlation structure in the clustered data, with clusters formed by the two ears of the same individual and the testing site and audiologist. We propose a linear mixed-effects model to take into account the multilevel correlation structures of the data. Both theoretically and in simulation studies, we compare single-ear linear regression models commonly used in published hearing loss studies with the proposed both-ears linear mixed models properly accounting for the multi-level correlations. Our findings include (1) when there are only participant-level covariates, the worse-ear linear regression models produce unbiased but typically less efficient estimators than the both-ear and average-ear approaches; (2) when there are ear-level confounders, the worse-ear method may lead to biased estimators and the average-ear method produces unbiased but typically less efficient estimators than the both-ear method; (3) the both-ear method may gain efficiency when additionally adjusting for testing sites and audiologists. As an illustrative example, we applied the single-ear and both-ear methods to assess aspirin-hearing association in the Nurses’ Health Study II.

1 |. INTRODUCTION

Hearing loss is a common and disabling global public health issue associated with social isolation, cognitive decline, depressive symptoms, and lower quality of life.^1–5 To understand the etiology of hearing loss, several epidemiological studies are providing much-needed population data. For example, the Conservation of Hearing Study (CHEARS) is a multicenter prospective study seeking to identify modifiable risk factors for acquired hearing loss. In our illustrative example, the CHEARS study population derives from participants in the ongoing longitudinal cohort Nurses Health Study (NHS) II. The Audiology Assessment Arm (AAA) within the NHS II CHEARS includes 3136 participants who underwent baseline and 3-year follow-up audiometric assessments performed by one of 71 certified audiologists at one of 32 test sites.

Correctly estimating the association between exposures and hearing loss from pure-tone audiometry data, such as the CHEARS-AAA data mentioned above, can be challenging due to the complex correlation structure. For instance, in the longitudinal setting, repeated measurements from multiple time points are correlated. In multicenter studies where subjects were tested at different sites by different audiologists, there are additional layers of intra-site and intra-audiologist correlations. Moreover, measurements from the same participant in the left ear and the right ear are paired. However, current analysis methods not only ignore the potential intra-site and intra-audiologist correlations in the pure-tone audiometry data, they often use the threshold measurements from a single ear such as the worse ear or better ear as the outcome.^6–9 In certain settings, this single-ear approach may result in biased association estimates, potentially resulting in misleading scientific conclusions. Moreover, while most published studies have tested participants’ hearing thresholds at multiple frequencies ranging from 0.25 to 8 kHz, the results are often summarized into a pure-tone average (PTA), which is calculated by averaging hearing threshold levels at a set of specified testing frequencies.^10–13 This common practice may also result in loss of information available in the individual frequency data.¹⁴ Overall, there is a need for reliable and efficient statistical methods that can address the complex correlation structure in hearing data and also fully utilize the measurement for each individual frequency.

There is a large literature on methods for longitudinal and correlated data.^15–17 The linear mixed effects models¹⁸ have become a popular tool for analyzing correlated data because of the flexibility in modeling the within-subject correlation and the availability of reliable and efficient software.^19,20 See Pinheiro²¹ for a review on linear mixed effects models for longitudinal data. Audiologists notice that hearing loss patterns vary across individuals, partially due to different patterns in aging among different organ systems.²² Therefore, the linear mixed effects models may be used to analyze hearing data, with individual variability characterized by subject-specific random effects.^22,23 Recently, concerning the commonly-made normality assumptions, Schielzeth et al²⁴ investigated the robustness issues regarding misspecification of the underlying distributions of the residual terms in the mixed effects models and found that model estimates are usually robust to violation of the distributional assumptions. Alternative analysis methods for correlated hearing data include the generalized estimating equations (GEEs)²⁵ that adjust for clusters as fixed effects and uncover the effect of interest from a population-average perspective.^13,26 While distributional assumptions are not required, the GEEs may not work well with large number of clusters but small cluster sizes.

In this paper, we present analytical methods that make maximum use of the data for estimating the associations between exposures and the continuous pure-tone audiometric hearing threshold measurements. Following a theoretical bias and efficiency evaluation of the commonly used single-ear methods, we emphasize the importance of the generalization from linear models to linear mixed-effects models in hearing studies and explicitly quantify the resulting bias and efficiency loss when failing to do so. We present a simulation study that compares results from analyses using linear mixed-effects models that account for multi-level correlations, models that do not account for multi-level correlations, and from linear regression methods based on single-ear data and assuming independence between participants. These methods are then illustrated in an analysis of the association between aspirin use and risk of hearing threshold elevation based on the CHEARS-AAA dataset.

2 |. MIXED-EFFECTS MODELS TAKING INTO ACCOUNT MULTI-LEVEL CLUSTERING

For the convenience of presentation, we use the context of hearing threshold data in AAA of CHEARS, where hearing thresholds were measured at two time points, the baseline and three-year follow up, to present our models. We first introduce notations that will be used throughout the paper. Suppose there are N study participants, N_a audiologists and N_s test sites in total. For each study participant, there are two audiometric assessments, at the baseline and the end of follow-up (3 years after the baseline assessment). The test sties at which the audiometry assessments were conducted and the audiologist(s) who conducted the assessments for participant i at baseline and follow-up are denoted as {s_i0, a_i0} and {s_i1, a_i1}, respectively. There are 4 levels of clusters: the ears $(j\in \{1,2\})$ are level 1 units, participants $(i\in \{1,2,\dots ,N\})$ are level 2 units, audiologists are level 3 units $\left(a_{i0},a_{i1}\in \left\{1,\dots ,N_a\right\}\right)$, and test sites are level 4 units $\left(s_{i0},s_{i1}\in \left\{1,2,\dots ,N_s\right\}\right)$. The ears are nested within subjects. In CHEARS-AAA, some audiologists work in multiple test sites, so audiologists are not nested within test sites. In a study where different sites do not share audiologists, audiologists would be nested within test sites at each time point.

First, assume that the hearing threshold measurements are available under only one frequency. The outcome of interest Y_ij is the change from baseline to the end of follow-up in the hearing threshold for the ith participant in the jth ear. Let X denote the participant-level exposure of interest (for example, aspirin use), W a participant-level confounder (for example, age at baseline), and Z an ear-level confounder (for example, hearing threshold measured at baseline). Each of X, W, Z can be either a scalar or a column vector. A linear mixed-effects model for Y_ij, with random effects for multilevel clusters, can be written as

$$\mathit{\text{Base model:}}{Y}_{ij}={\beta }_0+{\mathit{\beta }}_1{\mathbf{X}}_i+{\mathit{\beta }}_2{\mathbf{W}}_i+{\mathit{\beta }}_3{\mathbf{Z}}_{ij}+b_{s_{i0}}^{(site\text{0)}}+b_{s_{i1}}^{(site1)}+b_{a_{i0}}^{(aud\text{0)}}+b_{a_{i1}}^{(aud1)}+b_i^{(ind)}+{ϵ}_{ij},$$ (1)

for $i=1,\dots ,N$ and $j=1,2$, where β₁, β₂, and β₃ are row vectors of regression coefficients. In addition, $\left\{b_{s_{i0}}^{(site0)},b_{s_{i1}}^{(site\text{1})}\right\}$ are random site effects for the site where the ith participant was tested at baseline and the end of follow-up, respectively, $\left\{b_{a_{i0}}^{(aud0)},b_{a_{i1}}^{(aud1)}\right\}$ are the random audiologist effects for the audiologist testing the ith participant at baseline and the end of follow-up, respectively, and $b_i^{(ind)}$ is a random participant effect. These random effects and the residual term ϵ_ij are mutually independent and follow zero-mean normal distributions; that is, $b_{s_{i0}}^{(site\text{0)}}\sim N\left(0,{\sigma }_{(site\text{0})}^2\right)$, $b_{s_{i1}}^{(site1)}\sim N\left(0,{\sigma }_{(site1)}^2\right)$, $b_{a_{i0}}^{(aud0)}N\left(0,{\sigma }_{(aud\text{0})}^2\right)$, $b_{a_{i1}}^{(aud1)}\sim N\left(0,{\sigma }_{(aud1)}^2\right)$, $b_i^{(ind)}\sim N\left(0,{\sigma }_{(ind)}^2\right)$ and ${ϵ}_{ijq}\sim N\left(0,{\sigma }_{ϵ}^2\right)$. Schielzeth et al²⁴ investigated the robustness issues of the linear mixed-effects models and found that they are relatively insensitive to violations of the distributional assumptions. We refer to this model as the base model.

In the base model above, the outcomes of different participants tested by different audiologists at different sites are independent. We have $\mathrm{var}\left(Y_{ij}\mid {\mathbf{X}}_i,{\mathbf{W}}_i,{\mathbf{Z}}_{ij}\right)={\sigma }_{\text{(}aud\text{0)}}^2+{\sigma }_{(aud\text{1})}^2+{\sigma }_{\text{(}site\text{0)}}^2+{\sigma }_{\text{(}site\text{1)}}^2+{\sigma }_{(ind)}^2+{\sigma }_e^2$, denoted as V. Meanwhile, the random effect terms allow for dependence between measurements from the same participant, audiologist, or site. Takethe random effect for audiologist at baseline, $b_{a_{i0}}^{(aud0)}$, as an example: it allows for dependence between measurements tested by the same audiologist at baseline. Thus, the adjusted intra-cluster correlation coefficient (ICC) between the outcomes of two participants tested by the same audiologist at only the baseline is ${\sigma }_{(aud\text{0)}}^2/V$. It increases to $\left({\sigma }_{(aud\text{0)}}^2+{\sigma }_{(aud1)}^2\right)/V$ if assuming they share the same audiologists both at baseline and at Year 3. Furthermore, if the testing sites are also the same between these two participants, at Year 3, but not at baseline, then the ICC between the outcomes of these participants is $\left({\sigma }_{(aud0)}^2+{\sigma }_{(aud\text{1})}^2+{\sigma }_{(site\text{1})}^2\right)/V$. The within-participant between-ear ICC is $\left({\sigma }_{(aud\text{0)}}^2+{\sigma }_{(aud1)}^2+{\sigma }_{(site\text{0)}}^2+{\sigma }_{(site1)}^2+{\sigma }_{(ind)}^2\right)/V$.

The exposure effects are the same across all the participants in the base model. This assumption can be relaxed by adding a possibly vector-valued random slope to the effects of exposures, allowing the exposure effects to be participant-specific. This leads to

$$Randomslopemodel:Y_{ij}={\beta }_0+\left({\mathit{\beta }}_1+{\mathbf{b}}_{2i}^{(ind)}\right){\mathbf{X}}_i+{\mathit{\beta }}_2{\mathbf{W}}_i+{\mathit{\beta }}_3{\mathbf{Z}}_{ij}+b_{s_{i0}}^{(site\text{0)}}+b_{s_{i1}}^{(site\text{1)}}+b_{a_{i0}}^{(aud\text{0)}}+b_{a_{i1}}^{(aud\text{1)}}+b_{1i}^{(ind)}+{ϵ}_{ij},$$ (2)

where ${\mathbf{b}}_{2i}^{(ind)}\sim N\left(\mathbf{0},{\text{Σ}}_{slope}\right)$ is the random slope for exposure at the participant level. The average effect of exposure, when averaged over the population of participants, is β₁.

The hearing threshold measurements in CHEARS-AAA are for seven frequencies, 0.5, 1, 2, 3, 4, 6, and 8 kHz. To include data from all the frequencies in the same model, the outcome of interest Y_ijq is now the change from baseline to the end of follow-up in the hearing threshold for the ith participant in the jth ear at the qth frequency. A flexible linear mixed-effects model for Y_ijq, with participant-specific covariates and random effects for multilevel clusters that allow frequency-specific fixed and random effects, can be written as

$$\begin{array}{l}Fullfrequencymodel:Y_{ijq}={\beta }_{0q}+{\mathit{\beta }}_{1q}{\mathbf{X}}_i+{\mathit{\beta }}_{2q}{\mathbf{W}}_i+{\mathit{\beta }}_{3q}{\mathbf{Z}}_{ij}+b_{s_{i0}}^{(site\text{0})}+b_{s_{i1}}^{(site\text{1})}\\ +b_{a_{i0}}^{(aud0)}+b_{a_{i1}}^{(aud1)}+b_i^{(ind)}+b_{q{s}_{i0}}^{(site0.f)}+b_{q{s}_{i1}}^{(\mathit{\text{site}}1.f)}+b_{q{a}_{i0}}^{(aud0.f)}+b_{q{a}_{i1}}^{(aud1.f)}+b_{iq}^{(ind.f)}+{ϵ}_{ijq},\end{array}$$

for $i=1,\dots ,N$, $j=1,2$, $q=1,\dots ,Q$, where $\left\{b_{q{s}_{i0}}^{(site0.f)},b_{q{s}_{i1}}^{(site1.f)}\right\}$ are the random effects for frequency q for the sites at which the ith participant is tested at baseline and the end of follow-up, respectively, $\left\{b_{q{a}_{i0}}^{(aud0.f)},b_{q{a}_{i1}}^{(aud1.f)}\right\}$ are the similarly defined random audiologist effects, and $b_{iq}^{(ind.f)}$ is a random participant effect for frequency q. In addition to the zero-mean normality assumptions for the random effect and residual terms in the base model, we have $b_{q{s}_{i0}}^{(site0.f)}\sim N\left(0,{\sigma }_{(site0.f)}^2\right)$, $b_{q{s}_{i1}}^{(site1.f)}\sim N\left(0,{\sigma }_{(site1.f)}^2\right)$, $b_{q{a}_{i0}}^{(aud0.f)}\sim N\left(0,{\sigma }_{(aud0.f)}^2\right)$, $b_{q{a}_{i1}}^{(aud1.f)}\sim N\left(0,{\sigma }_{(aud1.f)}^2\right)$, and $b_{iq}^{(ind.f)}\sim N\left(0,{\sigma }_{(ind.f)}^2\right)$. In the mixed model, we can allow within-person covariance between $b_{q{s}_{im}}^{(sitem.f)}$ and $b_{s_{im}}^{(sitem)}$, between $b_{q{a}_{im}}^{(audm.f)}$ and $b_{a_{im}}^{(audm)}$, for $m=0,1$, and between $b_{iq}^{(ind.f)}$ and $b_i^{(ind)}$, denoting as $co{v}_{(sitem)}$, $co{v}_{(audm)}$ and $co{v}_{(ind)}$, respectively.

In the full frequency model above, for example, including the random effect for audiologist-frequency interaction, $b_{q{a}_{i0}}^{(aud0.f)}$, allows for the correlations between measurements for the same frequency tested by the same audiologist at baseline to differ from the correlations between measurements for different frequencies tested by the same audiologist at baseline. Denote the adjusted total variance, $\mathrm{var}\left(Y_{ijq}\mid {\mathbf{X}}_i,{\mathbf{W}}_i,{\mathbf{Z}}_{ij}\right)={\sigma }_{\text{(}aud\text{0)}}^2+{\sigma }_{\text{(}aud\text{1)}}^2+{\sigma }_{\text{(}site\text{0)}}^2+{\sigma }_{\text{(}stte\text{1)})}^2+{\sigma }_{\text{(}ind\text{)}}^2+{\sigma }_{(aud0.f)}^2+{\sigma }_{(aud1.f)}^2+{\sigma }_{(site0.f)}^2+{\sigma }_{(site\text{1}.f)}^2+{\sigma }_{(ind.f)}^2+{\mathit{cov}}_{(site0)}+{\mathit{cov}}_{(site1)}+{\mathit{cov}}_{(aud0)}+{\mathit{cov}}_{(aud1)}+{\mathit{cov}}_{(ind)}+{\sigma }_{ϵ}^2$, as V_f. The adjusted ICC between the outcomes of two participants tested by the same audiologist at only the baseline is ${\sigma }_{(aud0)}^2/V_f$ if the outcomes are for different frequencies and $\left({\sigma }_{(aud0)}^2+{\sigma }_{(aud0.f)}^2\right)/V_f$ if they are for the same frequency. The within-participant between-ear ICC is $\left({\sigma }_{(aud0)}^2+{\sigma }_{(aud1)}^2+{\sigma }_{(site\text{0)}}^2+{\sigma }_{(site\text{1})}^2+{\sigma }_{(ind)}^2\right)/V_f$ for two outcomes at different frequencies and $({\sigma }_{(aud\text{0)}}^2+{\sigma }_{(aud1)}^2+{\sigma }_{(site\text{0)}}^2+{\sigma }_{(site1)}^2+{\sigma }_{(ind)}^2+{\sigma }_{(aud\text{1}.f\text{)}}^2+{\sigma }_{(site\text{0}.f\text{)}}^2+{\sigma }_{(site\text{l}.f)}^2+{\sigma }_{(ind.f)}^2+co{v}_{(site\text{0)}}+co{v}_{(site1)}+co{v}_{(aud0)}+co{v}_{(aud1)}+co{v}_{(ind)})/V_f$ for outcomes at the same frequency.

After allowing fixed effects to vary over different frequencies, it is often reasonable to assume that the random effects for participants, sites and audiologists do not change across frequencies; in this case we can exclude the random effects for participant/site/audiologist-frequency interactions from the full frequency model, leading to $Y_{ijq}={\beta }_{0q}+{\mathit{\beta }}_{1q}{\mathbf{X}}_i+{\mathit{\beta }}_{2q}{\mathbf{W}}_i+{\mathit{\beta }}_{3q}{\mathbf{Z}}_{ij}+b_{s_{i0}}^{\text{(}site\text{0)}}+b_{s_{i1}}^{\text{(}site\text{1)}}+b_{a_{i0}}^{(aud0)}+b_{a_{i1}}^{(aud1)}+b_{1i}^{(ind)}+{ϵ}_{ijq}$. We can often further assume that the coefficients of some elements of W and Z do not change over frequency. Suppose that the data consist of results obtained at two testing frequencies, 1 and 6 kHz. Let $I(q=2)=1$ if test frequency = 6 kHz and $I(q=2)=0$ otherwise. The full frequency model may be simplified as

$$\begin{array}{l}Simplefrequencymodel:Y_{ijq}={\beta }_0+{\mathit{\beta }}_1{\mathbf{X}}_i+{\mathit{\beta }}_2{\mathbf{W}}_i+{\beta }_{3}I(q=2)+{\mathit{\beta }}_{4}I(q=2)\times {\mathbf{X}}_i\\ +{\beta }_5{\mathbf{\text{Z}}}_{ij}+b_{s_{i0}}^{(site\text{0)}}+b_{s_{i1}}^{(site\text{1)}}+b_{a_{i0}}^{(aud\text{0)}}+b_{a_{i1}}^{(aud\text{1)}}+b_i^{(ind)}+{ϵ}_{ij},\end{array}$$

where exposure impacts hearing at 1 and 6 kHz differently, which is captured by the interaction term $I(q=2)\times \mathbf{X}$. The effect of exposure on hearing threshold is represented by β₁ at 1 kHz and (β₁ + β₄) at 6 kHz.

Sample SAS code for each of the models can be found in Section 1 of the Supplementary Material.

3 |. THEORETICAL EVALUATION OF POTENTIAL BIAS AND EFFICIENCY LOSS IN SINGLE-EAR METHODS

If a participant-level exposure (eg, aspirin intake) influences hearing, the hearing measurements in both ears contain information about the exposure effects. The models described above allow us to make full use of the information from both ears of each participant, and the regression coefficients of the exposures in these both-ears models represent the exposure-outcome relationship. On the other hand, the commonly-used single-ear methods include the worse-ear and better-ear methods. The former uses data from the ear showing the larger change in hearing threshold, that is, $Y_i^w=\max \left(Y_{i1},Y_{i2}\right)$, and the latter uses data from the ear showing the smaller change in hearing threshold, that is, $Y_i^b=\min \left(Y_{i1},Y_{i2}\right)$. Another single-ear method is the average-ear method, which used averaged data from both ears, that is, $Y_i^a=\frac{1}{2}\left(Y_{i1}+Y_{i2}\right)$. In this section, we theoretically evaluate the potential bias and efficiency loss of the estimators from the single-ear methods by comparing them to the true parameters defined in the both-ears models.

3.1 |. Models with only participant-level covariates

In the analysis of the single-ear methods, if there are no ear-level covariates, the base model becomes

$$\mathit{Single-ear\; mixed-effects\; model}:Y_i={\tilde{\beta }}_0+{\tilde{\mathit{\beta }}}_1{\mathbf{X}}_i+{\tilde{\mathit{\beta }}}_2{\mathbf{W}}_i+{\tilde{b}}_{s_{i0}}^{(site0)}+{\tilde{b}}_{s_{i1}}^{(site\text{1)}}+{\tilde{b}}_{a_{i0}}^{(aud\text{0)}}+{\tilde{b}}_{a_{i1}}^{(aud1)}+{\widetilde{ϵ}}_i,$$ (3)

where Y_i is one of $Y_i^w$, $Y_i^b$, and $Y_i^a$, depending on which single-ear method we use, and we put a tilde on top of each parameter to distinguish the parameters and random effects from those for the both-ears method in Section 2.

The single-ear methods used in many published hearing loss studies^{4,10,12,13,27,28} treat data from individuals as independent and do not take into account site- and audiologist-level clusters. Without adjusting for the audiologist and site effects, we have the following standard linear regression model:

$$\mathit{Single-ear\; linear\; regression\; model}:Y_i={\tilde{\beta }}_0+{\tilde{\mathit{\beta }}}_1{\mathbf{X}}_i+{\tilde{\mathit{\beta }}}_2{\mathbf{W}}_i+{\widetilde{ϵ}}_i.$$ (4)

We denote the vectors of the estimated intercept and regression coefficients obtained from this single-ear linear regression model analysis as ${\widehat{\mathit{\beta }}}^w$, ${\widehat{\mathit{\beta }}}^b$, and ${\widehat{\mathit{\beta }}}^a$, corresponding to the three ways of constructing the outcome variables described earlier. Assuming normality for the outcome Y, it follows that

$$E\left({\widehat{\mathit{\beta }}}^w\right)={\left({\beta }_0+{\tilde{\sigma }}_{ϵ}/\sqrt{\pi },{\mathit{\beta }}_1,{\mathit{\beta }}_2\right)}^{\prime },$$

$$E\left({\widehat{\mathit{\beta }}}^b\right)={\left({\beta }_0-{\tilde{\sigma }}_{ϵ}/\sqrt{\pi },{\mathit{\beta }}_1,{\mathit{\beta }}_2\right)}^{\prime },$$

$$E\left({\widehat{\mathit{\beta }}}^a\right)={\left({\beta }_0,{\mathit{\beta }}_1,{\mathit{\beta }}_2\right)}^{\prime }=\mathit{\beta },$$

where ${\widetilde{ϵ}}_i\sim N\left(0,{\tilde{\sigma }}_{ϵ}^2\right),\varphi (\cdot )$ is the probability density function for standard normal distribution, and β₀, β₁ and β₂ are the regression parameters in the base model in (1) and the random slope model in (2) when β₃ = 0 in both models. Note that, when β₃ = 0, the base model and random slope model have the same mean of $E\left(Y_{ij}\mid {\mathbf{X}}_i,{\mathbf{W}}_i\right)$ and thus, they have the same fixed-effect parameters. See Section 2.1 of the Supplementary Material for details. Therefore, ${\widehat{\mathit{\beta }}}^a$ is an unbiased estimator of β, whereas ${\widehat{\mathit{\beta }}}^w$ and ${\widehat{\mathit{\beta }}}^b$ are biased in the intercept term but unbiased in the coefficients of exposure X and confounder W. This result also holds for the estimators from the single-ear mixed-effects model in (3), which has the same expectation $E\left(Y_i\mid {\mathbf{X}}_i,{\mathbf{W}}_i\right)$ as the single-ear linear regression model in (4).

We next compare the efficiency between the methods when the both-ears model is

$$Y_{ij}={\beta }_0+{\mathit{\beta }}_1{\overline{\mathbf{X}}}_i+b_i^{(ind)}+{ϵ}_{ij},$$

where $b_i^{(\mathit{\text{ind}})}\sim N\left(0,{\sigma }_b^2\right)$, ${ϵ}_{ij}\sim N\left(0,{\sigma }_{ϵ}^2\right)$, and the single-ear model is the same except that it does not include the random effects. Here, ${\overline{\mathbf{X}}}_i$ represents the set of participant-level covariates, which is possibly a vector; for example, ${\overline{\mathbf{X}}}_i={\left({\mathbf{X}}_i^{\prime },{\mathbf{W}}_i^{\prime }\right)}^{\prime }$. It follows that the maximum likelihood estimator (MLE) of ${\widehat{\mathit{\beta }}}_1$ has the following variance,

$$\mathrm{var}\left({\widehat{\mathit{\beta }}}_1\right)=\left(c{\sigma }_e^2+{\sigma }_b^2\right){\left(\begin{array}{cc}N& {\sum }_{i=1}^N{\overline{\mathbf{X}}}_i^{\prime }\\ {\sum }_{i=1}^N{\overline{\mathbf{X}}}_i& {\sum }_{i=1}^N{\overline{\mathbf{X}}}_i{\overline{\mathbf{X}}}_i^{\prime }\end{array}\right)}^{-1},$$

where c = 1 for the worse-ear and better-ear methods and c = 0.5 for the both-ears and average-ear methods. See Section 3.1 of the Supplementary Material for details. Therefore, the better-ear and worse-ear methods lead to estimators with the same variance. The average-ear method, on the other hand, results in estimators that are more efficient than the better-ear and worse-ear methods. Furthermore, the average-ear method is equally efficient as the both-ears method.

3.2 |. Models with ear-level confounders

When the both-ears model also includes ear-level confounders (ie, when ${\mathit{\beta }}_3\ne \mathbf{0}$ in the base model and the random slope model), the outcomes for the two ears of the same person can have different means; that is, when ${\mathbf{Z}}_{i1}\ne {\mathbf{Z}}_{i2}$,

$$E\left(Y_{i1}\mid {\mathbf{X}}_i,{\mathbf{W}}_i,{\mathbf{Z}}_{i1}\right)={\beta }_0+{\mathit{\beta }}_1{\mathbf{X}}_i+{\mathit{\beta }}_2{\mathbf{W}}_i+{\mathit{\beta }}_3{\mathbf{Z}}_{i1},$$

$$E\left(Y_{i2}\mid {\mathbf{X}}_i,{\mathbf{W}}_i,{\mathbf{Z}}_{i2}\right)={\beta }_0+{\mathit{\beta }}_1{\mathbf{X}}_i+{\mathit{\beta }}_2{\mathbf{W}}_i+{\mathit{\beta }}_3{\mathbf{Z}}_{i2}.$$

For presentational convenience, assume that the ear-level confounder is the baseline threshold. For the worse-earmethod, the baseline threshold of the worse ear, $Z_i^w$, is defined as $Z_i^w=Z_{ik}$, where $k={\text{argmax}}_j\left(Y_{ij}\right)$, and the single-ear linear regression model becomes

$$\mathit{Worse-ear\; linear\; regression\; model\; with\; ear-level\; covariates}:Y_i^w={\tilde{\beta }}_0+{\tilde{\mathit{\beta }}}_1{\mathbf{X}}_i+{\tilde{\mathit{\beta }}}_2{\mathbf{W}}_i+{\tilde{\beta }}_3{Z}_i^w+{\widetilde{ϵ}}_i.$$ (5)

We denote the vectors of the estimated fixed-effect intercept and regression coefficients obtained from this single-ear regression analysis as ${\widehat{\mathit{\beta }}}^w$. Adjusting for site and audiologist effects on top of that yields the following mixed-effects model:

$$\begin{gathered} \mathit{Worse-ear\; mixed-effects\; model\; with\; ear-level\; covariates}: \\ {Y}_i^w={\tilde{\beta }}_0+{\tilde{\mathit{\beta }}}_1{\mathbf{X}}_i+{\tilde{\mathit{\beta }}}_2{\mathbf{W}}_i+{\tilde{\beta }}_3{Z}_i^w+{\tilde{b}}_{s_{i0}}^{(site0)}+{\tilde{b}}_{s_{i1}}^{(site\text{1})}+{\tilde{b}}_{a_{i0}}^{(aud0)}+{\tilde{b}}_{a_{i1}}^{(aud1)}+{\widetilde{ϵ}}_i. \end{gathered}$$ (6)

Similarly, the baseline threshold of the better ear $Z_i^b$ is defined as $Z_i^b=Z_{ik}$, where $k={\text{argmin}}_j\left(Y_{ij}\right)$. The average baseline threshold of the two ears is $Z_i^a=\frac{1}{2}\left(Z_{i1}+Z_{i2}\right)$. The single-ear linear regression and mixed-effects models can then be defined for the better-ear and average-ear methods as in (5–6).

We denote the vectors of estimated regression parameters from the better-ear and average-ear linear regression models as ${\widehat{\mathit{\beta }}}^b$ and ${\widehat{\mathit{\beta }}}^a$, respectively. As shown in Section 2.2 of the Supplementary Material, ${\widehat{\mathit{\beta }}}^W$ and ${\widehat{\mathit{\beta }}}^b$ do not have closed forms for their expectations and typically do not converge to $\mathit{\beta }=\left({\beta }_0,{\mathit{\beta }}_1,{\mathit{\beta }}_2,{\beta }_3\right)$. As for the average-ear approach, ${\widehat{\mathit{\beta }}}^a$ is an unbiased estimator of β; that is, $E\left({\widehat{\mathit{\beta }}}^a\right)=\mathit{\beta }$. This is true for both the single-ear mixed-effects model (eg, model 6) and the single-ear linear regression model (eg, model 5).

Despite this, we compare the both-ears method under the model

$$Y_{ij}={\beta }_0+{\mathit{\beta }}_1{\overline{\mathbf{X}}}_i+{\beta }_2{Z}_{ij}+b_i^{(ind)}+{ϵ}_{ij},$$

where ${\overline{\mathbf{X}}}_i$ can be a vector as mentioned above, with the average-ear method under the standard linear regression model. It can be shown that the ordinary least square estimator from the average-ear method is less efficient than the MLE from the both-ears method when there is only participant random effect included in the mixed-effects model. See Section 3.2 of the Supplementary Material for details.

Note that unlike our full frequency model and simple frequency model in Section 2, the current single-ear methods typically do not include data from multiple frequencies as a clustered outcome. Instead, the single-ear methods are implemented with respect to the ΔPTA values calculated by averaging the threshold changes among multiple frequencies separately for two ears; that is, ${\text{ΔPTA}}_{ij}=\frac{1}{Q}{\sum }_{q=1}^Q{Y}_{ijq}$, denoted as $\overline{Y_{ij}}$. The outcomes in the worse-ear, better-ear and average-ear methods are based on the data from the ear with the larger ΔPTA, the smaller ΔPTA, or the average of both ears, respectively. In other words, $Y_i^w=\max \left(\overline{Y_{i1}},\overline{Y_{i2}}\right)$, $Y_i^b=\min \left(\overline{Y_{i1}},\overline{Y_{i2}}\right)$, and $Y_i^a=\frac{1}{2}\left(\overline{Y_{i1}}+\overline{Y_{i2}}\right)$. In the simulation studies in Section 4, we show that the conventional single-ear methods based on ΔPTA would not be able to estimate exposure effects correctly if exposure-frequency interaction exists.

4 |. SIMULATION STUDIES

We conducted simulation studies to compare among the following methods: (1) the worse-ear method, (2) the average-ear method, (3) the both-ears method including the random participant effect, and (4) the both-ears method including random site, audiologist, and participant effects. The former two methods entail the use of linear regression models and treat data from each individual as independent, thus do not consider any clusters, whereas the last two methods use mixed-effect models that take these correlations into account.

4.1 |. Simulation set-up

We considered a sample size of N = 1000 and N = 400 with 1000 simulation replications. To investigate the flexibility of our proposed models on clusters, we set N_a = 5, 40, 100 with N_s = 5, 10, 20, respectively, so that there are one, four, and five audiologists at each test site, respectively. We assumed that participants were randomly assigned to their test sites at baseline and returned to the same test site for their follow-up assessments. For both the baseline and follow-up assessments, participants were randomly assigned to one of the audiologists at their test sites. For $i=1,\dots ,N$, we considered the following covariates

• A binary exposure X_i; for example, aspirin intake defined as 1 if A_i > c_A and 0 otherwise, where A_i is a continuous variable representing the dose of aspirin-intake for participant i and c_A is a presepcified cutoff point;

• A continuous participant-level confounder W_i; for example, age;

• A continuous ear-level confounder Z_i; for example, hearing threshold at baseline ${\mathbf{Z}}_i=\left(Z_{i1},Z_{i2}\right)$.

The covariates are generated from the following multivariate normal distribution

$${\left(A_i,W_i,Z_{i1},Z_{i2}\right)}^{\prime }\sim N\left(\mathit{\mu },{\mathbf{V}}^{\frac{1}{2}}\mathbf{R}{\mathbf{V}}^{\frac{1}{2}}\right),$$

where

$$\mathit{\mu }=\left(\begin{array}{c}0\\ 62\\ 10\\ 10\end{array}\right),\mathbf{V}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 25& 0& 0\\ 0& 0& 64& 0\\ 0& 0& 0& 64\end{array}\right),\mathbf{R}=\left(\begin{array}{cccc}1& 0.2& 0.06& 0.06\\ 0.2& 1& 0.15& 0.15\\ 0.06& 0.15& 1& 0.7\\ 0.06& 0.15& 0.7& 1\end{array}\right).$$

The mean, variance, and correlation coefficients were specified to mimic the CHEARS-AAA dataset. We set c_A = 0.385 so that about 35% of the participants are aspirin-users, consistent with the CHEARS population. We generated outcome data according to Models 1–4 defined below.

$$\begin{gathered} \mathit{Model\; 1–base\; model\; without\; ear-level\; covariates:}: \\ {Y}_{ij}={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+b_{s_i}^{(site)}+b_{a_{i0}}^{(aud\text{0)}}+b_{a_{i1}}^{(aud1)}+b_i^{(ind)}+{ϵ}_{ij}; \end{gathered}$$

$$\begin{gathered} \mathit{Model\; 2–random\; slope\; model\; without\; ear-level\; covariates:}: \\ {Y}_{ij}={\beta }_0+\left({\beta }_1+b_{2i}^{(ind)}\right)X_i+{\beta }_2{W}_i+b_{s_i}^{(site)}+b_{a_{i0}}^{(aud0)}+b_{a_{i1}}^{(aud1)}+b_{1i}^{(ind)}+{ϵ}_{ij}; \end{gathered}$$

$$\begin{gathered} \mathrm{Model\; 3–random\; slope\; model\; with\; ear-level\; covariates:}: \\ {Y}_{ij}={\beta }_0+\left({\beta }_1+b_{2i}^{(ind)}\right)X_i+{\beta }_2{W}_i+{\beta }_3{Z}_{ij}+b_{s_i}^{(site)}+b_{a_{i0}}^{(aud\text{0)}}+b_{a_{i1}}^{(aud1)}+b_{1i}^{(ind)}+{ϵ}_{ij}; \end{gathered}$$

$$\begin{gathered} \mathit{Model\; 4–simple\; frequency\; model\; without\; ear-level\; covariates}: \\ {Y}_{ijq}={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+{\beta }_{3}I(q=2)+{\beta }_{4}I(q=2)X_i+b_{s_i}^{(site)}+b_{a_{i0}}^{(aud\text{0)}}+b_{a_{i1}}^{(aud1)}+b_{1i}^{(ind)}+{ϵ}_{ij}, \end{gathered}$$

where $i=1,\dots ,N$, $j=1,2$, and $q=1,2$. The regression parameters were based on a preliminary analysis of the CHEARS-AAA data. In Models 1–3, the parameters were set to $\left({\beta }_0,{\beta }_1,{\beta }_2,{\beta }_3\right)=(-3,0.2,0.1,-0.15)$. In Mode l4, the parameters were set to $\left({\beta }_0,{\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4\right)=(-3,0.2,0.1,-2,0.4)$. The random effects were sampled from independent standard normal distributions and thus, were not related to exposures. The adjusted within-site ICC and within-audiologist ICC were about 0.2 and 0.4, respectively. Finally, concerning the robustness of the mixed-effects models to misspecifications of underlying data distributions, we also considered scenarios where the random effects and residuals were assumed to be normal in analyses, whereas in fact, they were generated from asymmetric standard Gumbel distributions with cumulative distribution function (c.d.f.) $F(x)=e^{-e^{-x}}$ for $x\in (-\infty ,\infty )$. See Figure 1 in the Supplementary Material for the plots of the c.d.f. and probability density function (p.d.f.) of this distribution.

4.2 |. Discreteness of the hearing threshold data

In hearing tests, to determine the lowest levels at which sounds are reliably heard by each ear, pure tones were played to participants with 5dB increments. Hence, the hearing thresholds recorded were discrete and multiples of five. Generating the outcomes Y_ij and baseline threshold Z_ij for each participant according to Models 1–4, we created the ‘recorded’ hearing threshold of subject i’s jth ear at baseline as ${\tilde{Z}}_{ij}=5\lceil Z_{ij}/5\rceil$, where $\lceil x\rceil$ denotes the ceiling function taking as input a real number x and mapping it to the least integer greater than or equal to x. The ‘recorded’ outcome, change in hearing threshold, was then ${\tilde{Y}}_{ij}=5\lceil \left(Z_{ij}+Y_{ij}\right)/5\rceil -{\tilde{Z}}_{ij}$.

4.3 |. Operating characteristics

We analysed the simulated hearing datasets using SAS PROC MIXED for the mixed-effects models and SAS PROC REG for regular linear regression models. See Section 1 of the Supplementary Material for sample codes. The following operating characteristics (OC) were calculated to compare the finite sample bias and efficiency of the resulting estimators: (1) Relative bias, calculated as $\left(\frac{1}{1000}{\sum }_{r=1}^{1000}{\widehat{\beta }}^{(r)}-\beta \right)/\beta$, where ${\widehat{\beta }}^{(r)}$ denotes the estimate for the true parameter β in the rth simulation replicate; (2) empirical standard error (SE), calculated as the sample standard deviation of $\left\{{\widehat{\beta }}^{(1)},{\widehat{\beta }}^{(2)},\cdots ,{\widehat{\beta }}^{(1000)}\right\}$; (3) estimated SE, calculated by averaging the model-based SE estimates in the SAS output over 1000 replicates; (4) coverage rate (CR) of 95% confidence interval (CI), calculated as $\frac{1}{1000}{\sum }_{r=1}^{1000}I\left(L{B}^{(r)}\le \beta \le U{B}^{(r)}\right)$, where LB^(r) and UB^(r) denote the estimated upper bound and lower bound of the 95% CI for β in the rth simulation replicate, respectively.

4.4 |. Results of the simulation studies

When the data-generating mechanism only contains participant-level covariates (Models 1 and 2), all the models including the single-ear methods produced satisfactory point estimates of β₁ and β₂ in terms of bias and CR of the 95% CI. The model-based SE estimates of the linear models were close to the corresponding sandwich estimators. See Table 1 for the scenario with N = 1000, N_a = 40, N_s = 10 and Tables S1-S5 in the Supplementary Material for the other combinations of N, N_s, and N_a. However, as showed by the empirical SEs, the worse-ear method was less efficient compared to the both-ears methods and the average-ear method had comparable efficiency to the both-ears method that does not include the random site and audiologist effects. Further including the random site and audiologist, the both-ears methods were about 40% more efficient than the both-ears methods without the random site and audiologist effects. This is not surprising as sites and audiologists were not related to the exposure or age in this simulation; ignoring them does not cause bias in the point estimates but including them leads to smaller residual variance and thus improved efficiency of the regression coefficient estimates. When the residual terms and random effects were misspecified, the relative biases of the estimates from the single-ear methods and the both-ear methods not adjusting for sites and audiologists were lessthan 6% when N = 1000; however, they increased to 10%−13% when N = 400. The both-ears method including random sites and audiologists effects, on the other hand, still maintained satisfactory performance. See Tables S6 and S7 in the Supplementary Material for details. Rounding the hearing outcomes to multiples of 5 gave similar results. See Table S22 in the Supplementary Material for details.

TABLE 1.

Performances of different methods when data is simulated based on Models 1 and 2 (without ear-level covariates)

Parameter		Both-ears Aa	Both-ears Bb	Worse-earc	Average-eard
β1 (truth = 0.2) (Model 1)	Relative bias	0.87%	0.25%	−0.052%	0.25%
	Empirical SE	0.0841	0.139	0.143	0.139
	Estimated SE	0.0851	0.140	0.143	0.140
	CR of 95% CI	95.3%	94.5%	94.9%	94.6%
β2 (truth = 0.1) (Model 1)	Relative bias	−0.27%	−0.14%	−0.11%	−0.14%
	Empirical SE	0.00799	0.0134	0.0138	0.0134
	Estimated SE	0.00813	0.0134	0.0136	0.0134
	CR of 95% CI	95.0%	96.0%	95.6%	96.0%
β1 (truth = 0.2) (Model 2)	Relative bias	1.19%	0.55%	0.23%	0.53%
	Empirical SE	0.0985	0.148	0.152	0.149
	Estimated SE	0.101	0.150	0.148	0.145
	CR of 95% CI	95.6%	94.8%	94.4%	93.9%
β2 (truth = 0.1) (Model 2)	Relative bias	−0.31%	−0.19%	−0.13%	−0.16%
	Empirical SE	0.00872	0.0140	0.0144	0.0141
	Estimated SE	0.00876	0.0138	0.0142	0.0139
	CR of 95% CI	94.9%	95.6%	94.6%	95.6%

Note: The upper half displays the results when data is generated from Model 1 (the base model) and the lower half displays those when data is generated from Model 2 (the random slope model). Sample size N = 1000, with four audiologists at each of the ten sites.

^aBoth-ears method taking into account participants, sites, and audiologists-formed clusters, based on mixed-effects model $E\left(Y_{ij}\mid X_i,W_i,b_{s_i}^{(site)},b_{a_{i0}}^{(aud\text{0)}},b_{a_{i1}}^{(aud\text{1)}},b_i^{(ind)}\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+b_{s_i}^{(site)}+b_{a_{i0}}^{(aud0)}+b_{a_{i1}}^{(aud1)}+b_i^{(ind)}$ or $E\left(Y_{ij}\mid X_i,W_i,b_{s_i}^{(site)},b_{a_{i0}}^{(aud0)},b_{a_{i1}}^{(aud\text{1})},b_{1i}^{(ind)},b_{2i}^{(ind)}\right)={\beta }_0+\left({\beta }_1+b_{2i}^{(ind)}\right)X_i+{\beta }_2{W}_i+b_{s_i}^{(site)}+b_{a_{i0}}^{(aud0)}+b_{a_{i1}}^{(aud\text{1)}}+b_{1i}^{(ind)}$.

^bBoth-ears method ignoring sites and audiologists, based on mixed-effects model $E\left(Y_{ij}\mid X_i,W_i,b_i^{(ind)}\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+b_i^{(ind)}$ or $E\left(Y_{ij}\mid X_i,W_i,b_{1i}^{(ind)},b_{2i}^{(ind)}\right)={\beta }_0+\left({\beta }_1+b_{2i}^{(ind)}\right)X_i+{\beta }_2{W}_i+b_{1i}^{(ind)}$.

^cWorse-ear method ignoring sites and audiologists, based on linear regression model $E\left(Y_i^w\mid X_i,W_i\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i,Y_i^w=\max \left(Y_{i1},Y_{i2}\right)$.

^dAverage-ear method ignoring sites and audiologists, based on linear regression model $E\left(Y_i^a\mid X_i,W_i\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i,Y_i^a=\left(Y_{i1}+Y_{i2}\right)/2$.

When the data-generating mechanism contains an extra ear-specific confounder (Model 3), omitting the ear-specific covariate (in our case, the baseline threshold) in the worse-ear and average-ear linear regression resulted in parameter estimates with relative biases between −27% and −35%; see Table 2 for the scenario with N = 1000, N_a = 40, N_s = 10 and Tables S8-S12 in the Supplementary Material for the other combinations of N, N_s, and N_a. We also examined the performances of the single-ear and both-ears methods when the ear-specific covariate was included in the model. The outcome and baseline hearing threshold measurements in the worse-ear approach are those from the worse ear, that is,

$$E\left(Y_i^w\mid X_i,W_i{,\text{Baseline Threshold}}_i^w\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+{\beta }_3{\text{Baseline Threshold}}_i^w.$$

Our theoretical evaluation in Section 3 showed that this worse-ear approach can lead to biased estimates of regression coefficients. Both our theoretical evaluation (Section 2.2 of the Supplementary Material) and simulation studies suggest that the bias could be reduced by using the average baseline threshold as in the following model,

$$E\left(Y_i^w\mid X_i,W_i{,\text{Baseline Threshold}}_i^a\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+{\beta }_3{\text{Baseline Threshold}}_i^a.$$

See the comparison between the columns Worse-ear A and Worse-ear B in Table 2 and Tables S8-S12 in the Supplementary Material for details. Both the average-ear approach and the both-ears approach yielded satisfactory parameter estimates; their SE estimates were also comparable, and accounting for the random site and audiologist effects in the both-ears method further led to about 35% efficiency gains (first two columns in Table 2 and Tables S8-S12 in the Supplementary Material). Finally, relative biases increased under misspecification of residuals terms and random effects for all methods except the both-ears method including random sites and audiologists effects. See Tables S13 and S14 in the Supplementary Material for details. Similar results were obtained when using the discretized hearing data. See Table S23 in the Supplementary Material for details.

TABLE 2.

Performances of different methods when data is simulated based on Model 3 (random slope model with ear-level covariates)

Parameter		Bothears Aa	Bothears Bb	Worse-ear, misspecifiedc	Average-ear, misspecifiedd	Worseear Ae	Worseear Bf	Averageearg
β1 (truth = 0.2) (Model 3)	Relative bias	1.10%	0.52%	−28.00%	−28.70%	−1.89 %	1.01%	0.37%
	Empirical SE	0.0985	0.148	0.173	0.168	0.153	0.155	0.149
	Estimated SE	0.101	0.150	0.166	0.163	0.150	0.150	0.146
	CR of 95% CI	95.6%	94.6%	92.7%	92.5%	94.8%	94.7%	93.8%
β2 (truth = 0.1) (Model 3)	Relative bias	−0.33%	−0.19%	−34.90%	−35.00%	−2.66%	−0.18%	−0.19%
	Empirical SE	0.00880	0.0141	0.0162	0.0158	0.0147	0.0147	0.0142
	Estimated SE	0.00883	0.0139	0.0159	0.0156	0.0145	0.0145	0.0141
	CR of 95% CI	95.1%	95.4%	43.0%	41.5%	94.5%	95.2%	95.4%

Note: Sample size N = 1000, with four audiologists at each of the ten sites.

^aBoth-ears method taking into account participants, sites, and audiologists-formed clusters and ear-level covariates, based on mixed-effects model $E\left(Y_{ij}\mid X_i,W_i,Z_{ij},b_{s_i}^{(site)},b_{a_{i0}}^{(aud\text{0)})},b_{a_{i1}}^{(aud1)},b_{1i}^{(ind)},b_{2i}^{(ind)}\right)={\beta }_0+\left({\beta }_1+b_{2i}^{(ind)}\right)X_i+{\beta }_2{W}_i+{\beta }_3{Z}_{ij}+b_{s_i}^{(site)}+b_{a_{i0}}^{(aud0)}+b_{a_{i1}}^{(aud1)}+b_{1i}^{(ind)}$.

^bBoth-ears method taking into account ear-level covariates but ignoring sites and audiologists, based on mixed-effects model $E\left(Y_{ij}\mid X_i,W_i,Z_{ij},b_{1i}^{(ind)},b_{2i}^{(ind)}\right)={\beta }_0+\left({\beta }_1+b_{2i}^{(ind)}\right)X_i+{\beta }_2{W}_i+{\beta }_3{Z}_{ij}+b_{1i}^{(ind)}$.

^cWorse-ear method ignoring sites, audiologists and ear-level covariates, based on linear regression model $E\left(Y_i^w\mid X_i,W_i\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i,Y_i^w=\max \left(Y_{i1},Y_{i2}\right)$.

^dAverage-ear method ignoring sites, audiologists and ear-level covariates, based on linear regression model $E\left(Y_i^a\mid X_i,W_i\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i,Y_i^a=\left(Y_{i1}+Y_{i2}\right)/2$.

^eWorse-ear method taking into account ear-level covariates but ignoring sites and audiologists, based on linear regression model $E\left(Y_i^w\mid X_i,W_i,Z_i^w\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+{\beta }_3{Z}_i^w,Y_i^w=\max \left(Y_{i1},Y_{i2}\right),Z_i^w=\max \left(Z_{i1},Z_{i2}\right)$.

^fWorse-ear method ignoring sites and audiologists, using averaged ear-level covariates, based on linear regression model $E\left(Y_i^w\mid X_i,W_i,Z_i^a\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+{\beta }_3{Z}_i^a,Y_i^w=\max \left(Y_{i1},Y_{i2}\right),Z_i^a=\left(Z_{i1}+Z_{i2}\right)/2$.

^gAverage-ear method taking into account ear-level covariates but ignoring sites and audiologists, based on linear regression model $E\left(Y_i^a\mid X_i,W_i,Z_i^a\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+{\beta }_3{Z}_i^a,Y_i^a=\left(Y_{i1}+Y_{i2}\right)/2,Z_i^a=\left(Z_{i1}+Z_{i2}\right)/2$.

We further examined the situation where the pure-tone audiometry data are available at multiple frequencies at which the covariate of interest impacts hearing loss differently (Table 3 for the scenario with N = 1000, N_a = 40, N_s = 10 and Tables S15-S19 in the Supplementary Material for the other combinations of N, N_s, and N_a.). We simulated datasets using a two-frequency model (Model 4), which can be easily extended to include more frequencies by adding the appropriate indicator variables and interaction terms. The data-generating mechanism assumed that the exposure impacts the hearing threshold at 6 kHz (β₁ + β₄ = 0.6) more than it does at 1 kHz (β₁ = 0.2). The parameter estimates obtained using the worse-ear and average-ear methods based on ΔPTA were close to the average of the effects at 1 and 6 kHz, and therefore, biased when compared to the true parameters. See the last two columns of Table 3 and Tables S15-S19 in the Supplementary Material for details. The both-ears mixed-effects model including random site, audiologist, and participant effects presented in the first column in Table 3 and Tables S15-S19 in the Supplementary Material yielded relative biases as small as −0.34% and the smallest standard error estimates. Omitting the random site and audiologist effects gave valid estimates despite larger standard error estimates as shown in the second column in Table 3 and Tables S15-S19 in the Supplementary Material. Similar results were obtained under misspecification of the residual terms and random effects (Tables S20-S21 in the Supplementary Material). Discretizing the audiometry data also gave similar results (Table S24 in the Supplementary Material).

TABLE 3.

Performances of different methods when data is simulated based on Model 4 (simple frequency model without ear-level covariates)

Parameter		Both-ears Aa	Both-ears Bb	Worse-ear ΔPTAc	Average-ear ΔPTAd
β1 (truth = 0.2) (Model 4)	Relative bias	0.41%	0.72%	101%	101%
	Empirical SE	0.0815	0.136	0.135	0.134
	Estimated SE	0.0846	0.140	0.137	0.136
	CR of 95% CI	95.8%	95.7%	69.8%	69.2%
β1 + β4 (truth = 0.6) (Model 4)	Relative bias	−0.0086%	0.096%	−33.10%	−33.20%
	Empirical SE	0.0829	0.139	0.135	0.134
	Estimated SE	0.0846	0.140	0.137	0.136
	CR of 95% CI	95.7%	95.3%	69.7%	69.2%
β2 (truth = 0.1) (Model 4)	Relative bias	0.026%	0.67%	0.71%	0.67%
	Empirical SE	0.00735	0.0127	0.0129	0.0127
	Estimated SE	0.00742	0.0130	0.0131	0.0130
	CR of 95% CI	95.1%	95.9%	95.6%	95.9%

Note: Sample size N = 1000, with four audiologists at each of the ten sites.

^aBoth-ears method taking into account participants, sites, and audiologists-formed clusters, based on the PTA mixed-effects model $E\left(Y_{ijq}\mid X_i,W_i,b_{s_i}^{(site)},b_{a_{i0}}^{(aud\text{0)}},b_{a_{i1}}^{(aud1)},b_i^{(ind)}\right)={\beta }_0+{\beta }_1{X}_i+W_i+{\beta }_{3}I(q=2)+{\beta }_{4}I(q=2)\times X_i+b_{s_i}^{(site)}+b_{a_{i0}}^{(aud\text{0)}}+b_{a_{i1}}^{(aud\text{1})}+b_i^{(ind)}$.

^bBoth-ears method ignoring sites and audiologists, based on the PTA mixed-effects model $E\left(Y_{ijq}\mid X_i,W_i,b_i^{(ind)}\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i+{\beta }_{3}I(q=2)+{\beta }_{4}I(q=2)\times X_i+b_i^{(ind)}$.

^cWorse-ear method ignoring sites and audiologists, based on the PTA linear regression model $E\left(Y_i^w\mid X_i,W_i\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i,Y_i^w=\max \left(Y_{i1},Y_{i2}\right)$ in terms of ΔPTA.

^dAverage-ear method ignoring sites and audiologists, based on the PTA linear regression model $E\left(Y_i^a\mid X_i,W_i\right)={\beta }_0+{\beta }_1{X}_i+{\beta }_2{W}_i,Y_i^a=\left(Y_{i1}+Y_{i2}\right)/2$ in terms of ΔPTA.

In summary, if there are no ear-level confounders, the worse-ear method resulted in less efficient parameter estimates with small relative biases and satisfactory coverage rates. The average-ear method was as efficient as the both-ear method. However, if there are ear-level confounders, parameter estimates resulting from the worse-ear approach had larger relative biases. The average-ear method led to valid estimates which was slightly less efficient than the both-ear method. Note that, when the outcome is the change in the hearing threshold from baseline, the ear-level hearing threshold measured at baseline is likely to be confounders and should be adjusted for in the model. In our simulation setting, site and audiologist formed clusters in the outcome but was independent from the exposure. In this setting, ignoring site or audiologist in the model caused efficiency loss but not bias in the point estimates. If sites and audiologists are also related to exposure and age, omitting them in the model may lead to biased point estimates. In addition, when data are available at multiple frequencies with different exposure effects, the conventional methods based on the PTA led to parameter estimates with substantial relative biases, and the both-ears method allowing exposure-frequency interactions resulted valid estimates. Finally, while relative biases tended to rise under misspecification of the residual terms and random effects, the both-ears method including sites and audiologists effects gave satisfactory performance and was consistently better than other methods.

5 |. ILLUSTRATIVE EXAMPLE: ANALYSIS OF THE CHEARS-AAA DATA

The multicenter prospective study CHEARS and the study population are described in the introduction section briefly, with more details including ascertainment of covariates available in Curhan et al²⁹ We evaluated the aspirin-hearing threshold association using both our proposed both-ears mixed-effects models and the single-ear linear regression models that treat data from individuals as independent without taking into account any clusters. The CHEARS-AAA dataset consists of audiometric data at multiple test frequencies. In all models, aspirin exposure is binary indicating whether a participant is ever an aspirin user over the previous two years, whereas age and baseline pure-tone thresholds are continuous variables. Other potential confounders that are not explicitly listed in the regression models include smoking (categorical with never, past, current), body mass index (categorical with <25, 25–29, 30–34, 35–39, >40), physical activity (quintiles), hypertention (binary), diabetes (binary), dietary Approaches to Stop Hypertension (DASH) score (quintiles), total energy intake (continuous), acetaminophen use (categorical with none, 1 day/week, 2–3 days/week, 4–5 days/week, ≥6 days/week), ibuprofen use (categorical with none, 1 day/week, 2–3 days/week, 4–5 days/week, ≥6 days/week), occupational or leisure-time noise exposure (≥3 hours/week during any decade or not), and impulse or gunfire noise exposure (≥3 times/year during any decade or not).

We first analyze the data using the both-ears mixed-effects model

$$\begin{gathered} E\left[Y_{ijq}\mid X_{ij},b_{s_{i0}}^{(site\text{0)}},b_{s_{i1}}^{(site\text{1)}},b_{a_{i0}}^{(aud0)},b_{a_{i1}}^{(aud\text{1)}},b_i^{(ind)}\right]={\beta }_0+{\beta }_1{\text{Aspirin}}_i+{\beta }_2{\text{Age}}_i+{\beta }_3{\text{Baseline Threshold}}_{ijq} \\ +{\beta }_{4}I(\text{freq}=0.5\text{kHz})+\cdots +{\beta }_{9}I(\text{freq}=6\text{kHz})+{\beta }_{10}{\text{Aspirin}}_i\times I(\text{freq}=0.5\text{kHz})+\cdots +{\beta }_{15}{\text{Aspirin}}_i\times I(\text{freq}=6\text{kHz}) \\ +\cdots +b_{s_{i0}}^{(site\text{0)}}+b_{s_{i1}}^{(site1)}+b_{a_{i0}}^{(\mathit{\text{aud}}0)}+b_{a_{i1}}^{(aud1)}+b_i^{(ind)}, \end{gathered}$$

where Y_ijq is the 3-year change in the hearing threshold for participant i = 1, …, 3136 in ear j = 1, 2, at frequencies 0.5, 1, 2, 3, 4, 6, and 8 kHz (corresponding to q = 1,· · ·, 7, respectively). The inclusion of the frequency-aspirin and frequency-confounders interactions allows us to estimate the effect of aspirin use on hearing at different frequencies. The results for the high frequencies (6 and 8 kHz) are reported in Table 4, where the P-values are from the Wald test. As shown in Table 4, aspirin exposure is associated with an estimated increase of 0.727 dB (P = 0.0005, 95% CI: [0.317, 1.137]) in the elevation of the hearing threshold at 6 kHz after a 3-year follow-up. The results of the analyses for all the frequencies can be found in the top half of Table S25 in the Supplementary Material.

TABLE 4.

Results of the analyses of the CHEARS-AAA data at high frequencies using two-ear mixed-effects models and ordinary least square linear regression models

Model	Frequency	Estimate (SE)	P-value
Both-ears mixed-effects models with frequency-specific effectsa	6 kHz	0.727 (0.209)	0.0005
	8 kHz	0.388 (0.209)	0.0630
Both-ears frequency-specific mixed-effects modelsb	6 kHz	0.685 (0.272)	0.0119
	8 kHz	0.399 (0.297)	0.1799
Worse-ear linear regression modelsc	6 kHz	0.533 (0.328)	0.1041
	8 kHz	0.406 (0.349)	0.2458
Better-ear linear regression modelsd	6 kHz	0.758 (0.289)	0.0087
	8 kHz	0.283 (0.315)	0.3691
Average-ear linear regression modelse	6 kHz	0.630 (0.278)	0.0236
	8 kHz	0.323 (0.302)	0.2849

^aBoth-ears method taking into account participants, sites, and audiologists-formed clusters, based on mixed-effects model with frequency-specific effects for all the fixed effects covariates $E\left[Y_{ijq}\mid X_{ij},b_{s_{i0}}^{(site\text{0)}},b_{s_{i1}}^{(site1)}{b}_{a_{i0}}^{(aud\text{0)}},b_{a_{i1}}^{(aud1)},b^{(ind)}\right]={\beta }_0+{\beta }_1{\text{Aspirin}}_i+{\beta }_2{\text{Age}}_i+{\beta }_3{\text{Baseline Threshold}}_{ijq}+{\beta }_{4}I(\mathit{\text{freq}}=0.5\text{kHz})+\cdots +{\beta }_{9}I(\mathit{\text{freq}}=6\text{kHz})+{\beta }_{10}{\text{Aspirin}}_i\times I(\mathit{\text{freq}}=0.5\text{kHz})+\cdots +{\beta }_{15}{\mathit{\text{Aspirin}}}_i\times I(\mathit{\text{freq}}=6\text{kHz})+\cdots +b_{s_{i0}}^{(site0)}+b_{s_{i1}}^{(site\text{1)}}+b_{a_{i0}}^{(aud\text{0)}}+b_{a_{i1}}^{(aud\text{1)}}+b_i^{(ind)}$.

^bBoth-ears method taking into account participants, sites, and audiologists-formed clusters, based on frequency-specific mixed-effects model $E\left[Y_{ijq}\mid X_{ij},b_{s_{i0}}^{(site\text{0}.q\text{)}},b_{s_{i1}}^{(site\text{l}.q\text{)}},b_{a_{i0}}^{(aud0.q)},b_{a_{i1}}^{(aud1.q)},b_i^{(ind.q\text{)}}\right]={\beta }_{0q}+{\beta }_{1q}{\text{Aspirin}}_i+{\beta }_{2q}{\text{Age}}_i+{\beta }_{3q}{\text{Baseline Threshold}}_{ijq}+\cdots +b_{s_{i0}}^{(site\text{0}.q)}+b_{s_{i1}}^{(site1.q)}+b_{a_{i0}}^{(aud0.q)}+b_{a_{i1}}^{(aud\text{1}.q\text{)}}+b_i^{(ind.q\text{)}}\text{for}q=1,\dots ,7$.

^cWorse-ear method ignoring sites and audiologists, based on the frequency-specific linear regression model $E\left[Y_{iq}^w\right]={\beta }_{0q}+{\beta }_{1q}{X}_i+{\beta }_{2q}{W}_i+{\beta }_{3q}{\text{Baseline Threshold}}_{iq}^w+\dots \text{for}q=1,\dots ,10.Y_{iq}^w=\max \left(Y_{i1q},Y_{i2q}\right)$. ${\text{Baseline Threshold}}_{iq}^w={\text{Baseline Threshold}}_{ikq},\text{in which}k=\arg {\max }_j\left(Y_{ijq}\right)$.

^dBetter-ear method ignoring sites and audiologists, based on the frequency-specific linear regression model $E\left[Y_{iq}^b\right]={\beta }_{0q}+{\beta }_{1q}{X}_i+{\beta }_{2q}{W}_i+{\beta }_{3q}{\text{Baseline Threshold}}_{iq}^b+\dots \text{for}q=1,\dots ,10.Y_{iq}^b=\min \left(Y_{i1q},Y_{i2q}\right)$. ${\text{Baseline Threshold}}_{iq}^b={\text{Baseline Threshold}}_{ikq},\text{in which}k=\arg {\min }_j\left(Y_{ijq}\right)$.

^eAverage-ear method ignoring sites and audiologists, based on the frequency-specific linear regression model $E\left[Y_{iq}^a\right]={\beta }_{0q}+{\beta }_{1q}{X}_i+{\beta }_{2q}{W}_i+{\beta }_{3q}{\text{Baseline Threshold}}_{iq}^a+\dots \text{for}q=1,\dots ,10.Y_{iq}^a=\left(Y_{i1q}+Y_{i2q}\right)/2$. ${\text{Baseline Threshold}}_{iq}^a=\left({\text{Baseline Threshold}}_{i1q}+{\text{Baseline Threshold}}_{i2q}\right)/2.$

In addition, we fit separate mixed-effects models for each frequency,

$$\begin{gathered} E\left[Y_{ijq}\mid X_{ij},b_{s_{i0}}^{(site0.q)},b_{s_{i1}}^{(site1.q\text{)}},b_{a_{i0}}^{(aud0.q)},b_{a_{i1}}^{(aud\text{1}.q\text{)}},b_i^{(ind.q\text{)}}\right]={\beta }_{0q}+{\beta }_{1q}{\text{Aspirin}}_i+{\beta }_{2q}{\text{Age}}_i \\ +{\beta }_{3q}{\text{Baseline Threshold}}_{ijq}+\cdots +b_{s_{i0}}^{(site\text{0}.q)}+b_{s_{i1}}^{(site1.q)}+b_{a_{i0}}^{(aud\text{0}.q\text{)}}+b_{a_{i1}}^{(aud1.q)}+b_i^{\left(ind.q\right)}, \end{gathered}$$

for q = 1, …, 7. Aspirin exposure is associated with an estimated increase of 0.685 dB (P = 0.0119, 95% CI: [0.152, 1.218]) in the elevation of the hearing threshold at 6 kHz after a 3-year follow-up. See Table 4 and Table S25 in the Supplementary Material.

For comparison, we also fit single linear regression models where the outcome Y_iq is defined as the change in the hearing threshold of the worse ear (ie, ear with the larger increase in the threshold), the better ear (ie, ear with the smaller increase in the threshold), or the average change in the hearing threshold from two ears for participant i at frequency q. The model is expressed as follows, with some confounders introduced at the beginning of the section omitted.

$$\text{E}\left[Y_{iq}\right]={\beta }_{0q}+{\beta }_{1q}{\text{Aspirin}}_i+{\beta }_{2q}{\text{Age}}_i+{\beta }_{3q}{\text{Baseline Threshold}}_{iq}+\cdots$$

for q = 1, …, 10, where q = 1, …, 7 correspond to the frequencies 0.5 kHz , …, 8 kHz as in the mixed-effects models, and q = 8,9,10 correspond to the PTAs of low-frequency (LPTA_0.5,1,2kHz), mid-frequency (MPTA_3,4kHz), and high-frequency (HPTA_6,8kHz), respectively. The baseline pure-tone threshold is defined correspondingly, according to which ear was used to define Y_iq (see footnotes in Table 4). Following the better-ear approach, aspirin exposure is associated with an estimated increase of 0.758 dB (P = 0.0087, 95% CI: [0.192, 1.325]) in the elevation of the hearing threshold at 6 kHz after a 3-year follow-up. Following the average-ear approach, aspirin exposure is associated with an estimated increase of 0.630 dB (P = 0.0236, 95% CI: [0.0845, 1.175]) in the elevation of the hearing threshold at 6 kHz after a 3-year follow-up. The effect of aspirin exposure is not detected as significant following the worse-ear approach. See Table S26 of the Supplementary Material for the results of the analyses for all the frequencies.

6 |. DISCUSSION

In this article, we compare, both theoretically and in simulation studies, the commonly used existing methods in hearing-loss literature, which utilize single-ear data and treat each participant as independent, to proposed both-ears methods that are based on linear mixed-effects models. We found that (1) the worse-ear and better-ear methods may lead to less efficient estimators than the both-ears and average-ear methods if there are no ear-level confounders, and they may lead to biased estimates when there exist ear-level confounders; (2) the average-ear method is valid whether or not there are ear-level confounders; in terms of efficiency, it is comparable to the both-ear method that includes a random individual effect to take into account the between-ear correlation if there are no ear-level confounders; however, if there exist ear-level confounders, the average-ear method may lead to less efficient estimators than the both-ears method; (3) the both-ear methods can gain efficiency by including site and audiologist random effects if site and audiologist are correlated with the hearing outcome but not the exposure.

This article focuses on the situations when sites and audiologists are correlated only with the hearing outcome. In the cases when sites and audiologists are confounders, they should be included in the model; not including them may cause bias in effect estimates.³⁰ Other than the confounding issue, we have assumed in our study that the random effects follow normal distributions. However, according to our simulation studies in Section 4, the both-ears methods including random sites and audiologists effects maintained the best performances when such assumptions were violated. Our proposed both-ears mixed effects models are expected to work in settings of a typical hearing study. The full frequency model includes the audiologist-frequency, site-frequency, and participant-frequency interactions. Additional random intercepts and slopes can be similarly specified to take into account other interaction effects. According to the simulation studies in Section 4, the SAS program works well with a low computational burden across simulation settings with flexible sample sizes, cluster numbers and cluster sizes that are realistic in hearing studies. Note that, if the between-cluster variation is close to zero, the estimated variance of the random effect for the cluster could be negative.^31–33 While we did not encounter this boundary issue in our settings, one could add restrictions as described in Thompson³⁴ to avoid obtaining estimates out of the parameter space. Additional investigations of such negative variance problems can be found in Fletcher and Underwood³⁵ and Leithy et al.³² Furthermore, we can perform statistical tests on zero variance components in linear mixed-effects models.^36,37 If the variance is zero, one can consider remove the random effect from the model.

Moreover, linear mixed effects models are advantageous in uncovering individual-specific effects. This is particularly desirable when dealing with correlated hearing data since hearing loss patterns are expected to vary across participants. Other existing statistical options such as marginal methods focus on the population-average effects. See Chen et al³⁸ for a both-ears method that analyzes hearing data based on GEEs without taking into account sites and audiologists effects. While mixed-effects models adjust for sites and audiologists using random effects, the GEE method can adjust for them using the fixed effects. Although the GEE method does not require any distributional assumptions, it may not be suitable forthescenarioswheremanysmallclustersareincludedasfixed-effects.Athoroughcomparisonofthemarginalmethods such as the GEE method in the application of hearing studies with mixed effects models is a topic of our future research.

Finally, we comment that the CHEARS-AAA example only has two time points. For studies with three or more time points, one could extend our proposed model by including time, time-exposure interactions, and time-confounders interactions as the fixed effects. As for the random effects in the models, one could consider the interactions between site/audiologist and time, similar to those between site/audiologist and frequency in the full frequency model in Section 2. In addition, taking the simple frequency model in Section 2 as an example, the participant random effect $b_i^{(ind)}$ can be expanded to $b_i^{(ind)}+b_{it(i)}^{(ind.time)}$, where the subscripts t(i) indicates that time points are nested within participants. The additional participant-time random interaction allows for the ICC of the outcomes of the same participant at the same time point to be larger than that of the same participant at distinct time points.

DATA AVAILABILITY STATEMENT

The data for the Audiology Assessment Arm are not publicly available.