Construction of hearing percentiles in women with non-constant variance from the linear mixed-effects model

Current age-specific reference standards for adult hearing thresholds are primarily cross-sectional in nature and vary in the degree of screening of the reference sample for noise-induced hearing loss and other hearing problems. We develop methods to construct age-specific percentiles for longitudinal data that have been modelled using the linear mixed-effects model. We apply these methods to construct percentiles of hearing level using data from a carefully screened sample of women from the Baltimore Longitudinal Study of Aging. However, the variation in the residuals and random effects from the linear mixed-effects model does not remain constant with age and frequency of the stimulus tone. In addition, the distribution of the hearing levels is not symmetric about the mean. We develop a number of methods to use the output from the linear mixed-effects model to construct percentiles that do not have constant variance. We use a transformation of the hearing levels to provide for skewness in the final percentile curves. The change in the variation of the residuals and random effects is modelled as a function of beginning age and frequency and we use this variance function to construct the hearing percentiles. We present a number of approaches. First, we use the absolute values of the population residuals to model the total deviation about the mean as a function of beginning age and frequency. Second, we model the standard deviation in the person-specific (cluster) residuals as well as the standard deviation in the estimated random effects. Finally, we use weighted least squares with the regressions on the absolute cluster residuals and absolute estimated random effects where the weights are the reciprocal of the standard deviations of their estimates.